1.3 The Derivative Of A Function At A Point - Active Calculus

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• Front Matter
  • Colophon
  • Acknowledgements
  • Active Calculus: Our Goals
  • Features of the Text
  • Students! Read this!
  • Instructors! Read this!
• 1 Understanding the Derivative
  • 1.1 How do we measure velocity?
    • 1.1.1 Introduction
      • 1.1.1  
    • 1.1.2 Position and average velocity
      • 1.1.2  
    • 1.1.3 Instantaneous Velocity
      • 1.1.3  
      • 1.1.3  
    • 1.1.4 Summary
    • 1.1.5 Exercises
  • 1.2 The notion of limit
    • 1.2.1 Introduction
      • 1.2.1  
    • 1.2.2 The Notion of Limit
      • 1.2.2  
    • 1.2.3 Instantaneous Velocity
      • 1.2.3  
      • 1.2.3  
    • 1.2.4 Summary
    • 1.2.5 Exercises
  • 1.3 The derivative of a function at a point
    • 1.3.1 Introduction
      • 1.3.1  
    • 1.3.2 The Derivative of a Function at a Point
      • 1.3.2  
      • 1.3.2  
      • 1.3.2  
    • 1.3.3 Summary
    • 1.3.4 Exercises
  • 1.4 The derivative function
    • 1.4.1 Introduction
      • 1.4.1  
    • 1.4.2 How the derivative is itself a function
      • 1.4.2  
      • 1.4.2  
    • 1.4.3 Summary
    • 1.4.4 Exercises
  • 1.5 Interpreting, estimating, and using the derivative
    • 1.5.1 Introduction
      • 1.5.1  
    • 1.5.2 Units of the derivative function
      • 1.5.2  
    • 1.5.3 Toward more accurate derivative estimates
      • 1.5.3  
      • 1.5.3  
    • 1.5.4 Summary
    • 1.5.5 Exercises
  • 1.6 The second derivative
    • 1.6.1 Introduction
      • 1.6.1  
    • 1.6.2 Increasing or decreasing
    • 1.6.3 The Second Derivative
    • 1.6.4 Concavity
      • 1.6.4  
      • 1.6.4  
      • 1.6.4  
    • 1.6.5 Summary
    • 1.6.6 Exercises
  • 1.7 Limits, continuity, and differentiability
    • 1.7.1 Introduction
      • 1.7.1  
    • 1.7.2 Having a limit at a point
      • 1.7.2  
    • 1.7.3 Being continuous at a point
      • 1.7.3  
    • 1.7.4 Being differentiable at a point
      • 1.7.4  
    • 1.7.5 Summary
    • 1.7.6 Exercises
  • 1.8 The tangent line approximation
    • 1.8.1 Introduction
      • 1.8.1  
    • 1.8.2 The tangent line
    • 1.8.3 The local linearization
      • 1.8.3  
      • 1.8.3  
      • 1.8.3  
    • 1.8.4 Summary
    • 1.8.5 Exercises
• 2 Computing Derivatives
  • 2.1 Elementary derivative rules
    • 2.1.1 Introduction
      • 2.1.1  
    • 2.1.2 Some Key Notation
    • 2.1.3 Constant, Power, and Exponential Functions
      • 2.1.3  
    • 2.1.4 Constant Multiples and Sums of Functions
      • 2.1.4  
      • 2.1.4  
    • 2.1.5 Summary
    • 2.1.6 Exercises
  • 2.2 The sine and cosine functions
    • 2.2.1 Introduction
      • 2.2.1  
    • 2.2.2 The sine and cosine functions
      • 2.2.2  
      • 2.2.2  
      • 2.2.2  
    • 2.2.3 Summary
    • 2.2.4 Exercises
  • 2.3 The product and quotient rules
    • 2.3.1 Introduction
      • 2.3.1  
    • 2.3.2 The product rule
      • 2.3.2  
    • 2.3.3 The quotient rule
      • 2.3.3  
    • 2.3.4 Combining rules
      • 2.3.4  
    • 2.3.5 Summary
    • 2.3.6 Exercises
  • 2.4 Derivatives of other trigonometric functions
    • 2.4.1 Introduction
      • 2.4.1  
    • 2.4.2 Derivatives of the cotangent, secant, and cosecant functions
      • 2.4.2  
      • 2.4.2  
      • 2.4.2  
    • 2.4.3 Summary
    • 2.4.4 Exercises
  • 2.5 The chain rule
    • 2.5.1 Introduction
      • 2.5.1  
    • 2.5.2 The chain rule
      • 2.5.2  
    • 2.5.3 Using multiple rules simultaneously
      • 2.5.3  
      • 2.5.3  
    • 2.5.4 The composite version of basic function rules
    • 2.5.5 Summary
    • 2.5.6 Exercises
  • 2.6 Derivatives of inverse functions
    • 2.6.1 Introduction
      • 2.6.1  
    • 2.6.2 Basic facts about inverse functions
    • 2.6.3 The derivative of the natural logarithm function
      • 2.6.3  
    • 2.6.4 Inverse trigonometric functions and their derivatives
      • 2.6.4  
      • 2.6.4  
    • 2.6.5 The link between the derivative of a function and the derivative of its inverse
    • 2.6.6 Summary
    • 2.6.7 Exercises
  • 2.7 Derivatives of functions given implicitly
    • 2.7.1 Introduction
      • 2.7.1  
    • 2.7.2 Implicit Differentiation
      • 2.7.2  
      • 2.7.2  
      • 2.7.2  
    • 2.7.3 Summary
    • 2.7.4 Exercises
• 3 Using Derivatives
  • 3.1 Related rates
    • 3.1.1 Introduction
      • 3.1.1  
    • 3.1.2 Related Rates Problems
      • 3.1.2  
      • 3.1.2  
      • 3.1.2  
      • 3.1.2  
    • 3.1.3 Summary
    • 3.1.4 Exercises
  • 3.2 Using derivatives to evaluate limits
    • 3.2.1 Introduction
      • 3.2.1  
    • 3.2.2 Using derivatives to evaluate indeterminate limits of the form \(\frac{0}{0}\text{.}\)
      • 3.2.2  
      • 3.2.2  
    • 3.2.3 Limits involving \(\infty\)
      • 3.2.3  
    • 3.2.4 Summary
    • 3.2.5 Exercises
  • 3.3 Using derivatives to identify extreme values
    • 3.3.1 Introduction
      • 3.3.1  
    • 3.3.2 Critical numbers and the first derivative test
      • 3.3.2  
    • 3.3.3 The second derivative test
      • 3.3.3  
      • 3.3.3  
    • 3.3.4 Summary
    • 3.3.5 Exercises
  • 3.4 Using derivatives to describe families of functions
    • 3.4.1 Introduction
      • 3.4.1  
    • 3.4.2 Describing families of functions in terms of parameters
      • 3.4.2  
      • 3.4.2  
      • 3.4.2  
    • 3.4.3 Summary
    • 3.4.4 Exercises
  • 3.5 Global optimization
    • 3.5.1 Introduction
      • 3.5.1  
    • 3.5.2 Global Optimization
      • 3.5.2  
      • 3.5.2  
    • 3.5.3 Moving toward applications
      • 3.5.3  
    • 3.5.4 Summary
    • 3.5.5 Exercises
  • 3.6 Applied optimization
    • 3.6.1 Introduction
      • 3.6.1  
    • 3.6.2 More applied optimization problems
      • 3.6.2  
      • 3.6.2  
      • 3.6.2  
      • 3.6.2  
    • 3.6.3 Summary
    • 3.6.4 Exercises
• 4 The Definite Integral
  • 4.1 Determining distance traveled from velocity
    • 4.1.1 Introduction
      • 4.1.1  
    • 4.1.2 Area under the graph of the velocity function
      • 4.1.2  
    • 4.1.3 Two approaches: area and antidifferentiation
      • 4.1.3  
    • 4.1.4 When velocity is negative
      • 4.1.4  
    • 4.1.5 Summary
    • 4.1.6 Exercises
  • 4.2 Riemann sums
    • 4.2.1 Introduction
      • 4.2.1  
    • 4.2.2 Sigma Notation
      • 4.2.2  
    • 4.2.3 Riemann Sums
      • 4.2.3  
    • 4.2.4 When the function is sometimes negative
      • 4.2.4  
    • 4.2.5 Summary
    • 4.2.6 Exercises
  • 4.3 The definite integral
    • 4.3.1 Introduction
      • 4.3.1  
    • 4.3.2 The definition of the definite integral
      • 4.3.2  
    • 4.3.3 Some properties of the definite integral
      • 4.3.3  
    • 4.3.4 How the definite integral is connected to a function’s average value
      • 4.3.4  
    • 4.3.5 Summary
    • 4.3.6 Exercises
  • 4.4 The Fundamental Theorem of Calculus
    • 4.4.1 Introduction
      • 4.4.1  
    • 4.4.2 The Fundamental Theorem of Calculus
      • 4.4.2  
    • 4.4.3 Basic antiderivatives
      • 4.4.3  
    • 4.4.4 The total change theorem
      • 4.4.4  
    • 4.4.5 Summary
    • 4.4.6 Exercises
• 5 Evaluating Integrals
  • 5.1 Constructing accurate graphs of antiderivatives
    • 5.1.1 Introduction
      • 5.1.1  
    • 5.1.2 Constructing the graph of an antiderivative
      • 5.1.2  
    • 5.1.3 Multiple antiderivatives of a single function
      • 5.1.3  
    • 5.1.4 Functions defined by integrals
      • 5.1.4  
    • 5.1.5 Summary
    • 5.1.6 Exercises
  • 5.2 The Second Fundamental Theorem of Calculus
    • 5.2.1 Introduction
      • 5.2.1  
    • 5.2.2 The Second Fundamental Theorem of Calculus
      • 5.2.2  
    • 5.2.3 Understanding Integral Functions
      • 5.2.3  
    • 5.2.4 Differentiating an Integral Function
      • 5.2.4  
    • 5.2.5 Summary
    • 5.2.6 Exercises
  • 5.3 Integration by substitution
    • 5.3.1 Introduction
      • 5.3.1  
    • 5.3.2 Reversing the Chain Rule: First Steps
      • 5.3.2  
    • 5.3.3 Reversing the Chain Rule: \(u\)-substitution
      • 5.3.3  
    • 5.3.4 Evaluating Definite Integrals via \(u\)-substitution
      • 5.3.4  
    • 5.3.5 Summary
    • 5.3.6 Exercises
  • 5.4 Integration by parts
    • 5.4.1 Introduction
      • 5.4.1  
    • 5.4.2 Reversing the Product Rule: Integration by Parts
      • 5.4.2  
    • 5.4.3 Some Subtleties with Integration by Parts
      • 5.4.3  
    • 5.4.4 Using Integration by Parts Multiple Times
      • 5.4.4  
    • 5.4.5 Evaluating Definite Integrals Using Integration by Parts
    • 5.4.6 When \(u\)-substitution and Integration by Parts Fail to Help
    • 5.4.7 Summary
    • 5.4.8 Exercises
  • 5.5 Other options for finding algebraic antiderivatives
    • 5.5.1 Introduction
      • 5.5.1  
    • 5.5.2 The Method of Partial Fractions
      • 5.5.2  
    • 5.5.3 Using an Integral Table
      • 5.5.3  
    • 5.5.4 Using Computer Algebra Systems
    • 5.5.5 Summary
    • 5.5.6 Exercises
  • 5.6 Numerical integration
    • 5.6.1 Introduction
      • 5.6.1  
    • 5.6.2 The Trapezoid Rule
      • 5.6.2  
    • 5.6.3 Comparing the Midpoint and Trapezoid Rules
    • 5.6.4 Simpson’s Rule
      • 5.6.4  
    • 5.6.5 Overall observations regarding \(L_n\text{,}\) \(R_n\text{,}\) \(T_n\text{,}\) \(M_n\text{,}\) and \(S_{2n}\text{.}\)
      • 5.6.5  
    • 5.6.6 Summary
    • 5.6.7 Exercises
• 6 Using Definite Integrals
  • 6.1 Using definite integrals to find area and length
    • 6.1.1 Introduction
      • 6.1.1  
    • 6.1.2 The Area Between Two Curves
      • 6.1.2  
    • 6.1.3 Finding Area with Horizontal Slices
      • 6.1.3  
    • 6.1.4 Finding the length of a curve
      • 6.1.4  
    • 6.1.5 Summary
    • 6.1.6 Exercises
  • 6.2 Using definite integrals to find volume
    • 6.2.1 Introduction
      • 6.2.1  
    • 6.2.2 The Volume of a Solid of Revolution
      • 6.2.2  
    • 6.2.3 Revolving about the \(y\)-axis
      • 6.2.3  
    • 6.2.4 Revolving about horizontal and vertical lines other than the coordinate axes
      • 6.2.4  
    • 6.2.5 Summary
    • 6.2.6 Exercises
  • 6.3 Density, mass, and center of mass
    • 6.3.1 Introduction
      • 6.3.1  
    • 6.3.2 Density
      • 6.3.2  
    • 6.3.3 Weighted Averages
      • 6.3.3  
    • 6.3.4 Center of Mass
      • 6.3.4  
    • 6.3.5 Summary
    • 6.3.6 Exercises
  • 6.4 Physics applications: work, force, and pressure
    • 6.4.1 Introduction
      • 6.4.1  
    • 6.4.2 Work
      • 6.4.2  
    • 6.4.3 Work: Pumping Liquid from a Tank
      • 6.4.3  
    • 6.4.4 Force due to Hydrostatic Pressure
      • 6.4.4  
    • 6.4.5 Summary
    • 6.4.6 Exercises
  • 6.5 Improper integrals
    • 6.5.1 Introduction
      • 6.5.1  
    • 6.5.2 Improper Integrals Involving Unbounded Intervals
      • 6.5.2  
    • 6.5.3 Convergence and Divergence
      • 6.5.3  
    • 6.5.4 Improper Integrals Involving Unbounded Integrands
      • 6.5.4  
    • 6.5.5 Summary
    • 6.5.6 Exercises
• 7 Differential Equations
  • 7.1 An introduction to differential equations
    • 7.1.1 Introduction
      • 7.1.1  
    • 7.1.2 What is a differential equation?
      • 7.1.2  
    • 7.1.3 Differential equations in the world around us
      • 7.1.3  
    • 7.1.4 Solving a differential equation
      • 7.1.4  
    • 7.1.5 Summary
    • 7.1.6 Exercises
  • 7.2 Qualitative behavior of solutions to differential equations
    • 7.2.1 Introduction
      • 7.2.1  
    • 7.2.2 Slope fields
      • 7.2.2  
    • 7.2.3 Equilibrium solutions and stability
      • 7.2.3  
    • 7.2.4 Summary
    • 7.2.5 Exercises
  • 7.3 Euler’s method
    • 7.3.1 Introduction
      • 7.3.1  
    • 7.3.2 Euler’s Method
      • 7.3.2  
      • 7.3.2  
    • 7.3.3 The error in Euler’s method
    • 7.3.4 Summary
    • 7.3.5 Exercises
  • 7.4 Separable differential equations
    • 7.4.1 Introduction
      • 7.4.1  
    • 7.4.2 Solving separable differential equations
      • 7.4.2  
      • 7.4.2  
      • 7.4.2  
    • 7.4.3 Summary
    • 7.4.4 Exercises
  • 7.5 Modeling with differential equations
    • 7.5.1 Introduction
      • 7.5.1  
    • 7.5.2 Developing a differential equation
      • 7.5.2  
      • 7.5.2  
    • 7.5.3 Summary
    • 7.5.4 Exercises
  • 7.6 Population growth and the logistic equation
    • 7.6.1 Introduction
      • 7.6.1  
    • 7.6.2 The earth’s population
      • 7.6.2  
    • 7.6.3 Solving the logistic differential equation
      • 7.6.3  
    • 7.6.4 Summary
    • 7.6.5 Exercises
• 8 Taylor Polynomials and Taylor Series
  • 8.1 Extending local linearization
    • 8.1.1 Introduction
      • 8.1.1  
    • 8.1.2 Finding a quadratic approximation
      • 8.1.2  
    • 8.1.3 Over and over again
      • 8.1.3  
    • 8.1.4 As the degree of the approximation increases
      • 8.1.4  
    • 8.1.5 Summary
    • 8.1.6 Exercises
  • 8.2 Taylor polynomials
    • 8.2.1 Introduction
      • 8.2.1  
    • 8.2.2 Taylor polynomials
      • 8.2.2  
    • 8.2.3 Taylor polynomial approximations centered at an arbitrary value \(a\)
      • 8.2.3  
      • 8.2.3  
    • 8.2.4 Summary
    • 8.2.5 Exercises
  • 8.3 Geometric sums
    • 8.3.1 Introduction
      • 8.3.1  
    • 8.3.2 Finite Geometric Series
      • 8.3.2  
    • 8.3.3 Infinite Geometric Series
      • 8.3.3  
    • 8.3.4 How geometric series naturally connect to Taylor polynomials
      • 8.3.4  
    • 8.3.5 Summary
    • 8.3.6 Exercises
  • 8.4 Taylor series
    • 8.4.1 Introduction
      • 8.4.1  
    • 8.4.2 Taylor series and the Ratio Test
      • 8.4.2  
    • 8.4.3 Taylor series of several important functions
      • 8.4.3  
    • 8.4.4 Summary
    • 8.4.5 Exercises
  • 8.5 Finding and using Taylor series
    • 8.5.1 Introduction
      • 8.5.1  
    • 8.5.2 Using substitution and algebra to find new Taylor series expressions
      • 8.5.2  
    • 8.5.3 Differentiating and integrating Taylor series
      • 8.5.3  
      • 8.5.3  
    • 8.5.4 Summary
    • 8.5.5 Exercises
  • 8.6 Quantifying the accuracy of approximations
    • 8.6.1 Introduction
      • 8.6.1  
    • 8.6.2 Alternating series of real numbers
      • 8.6.2  
    • 8.6.3 Error Approximations for Taylor Polynomials
      • 8.6.3  
    • 8.6.4 Summary
    • 8.6.5 Exercises
• Back Matter
  • A A Short Table of Integrals
  • B Answers to Activities
  • C Answers to Selected Exercises
  • Index
  • Colophon

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Section1.3The derivative of a function at a point

Motivating Questions

• How is the average rate of change of a function on a given interval defined, and what does this quantity measure?🔗 🔗
• How is the instantaneous rate of change of a function at a particular point defined? How is the instantaneous rate of change linked to average rate of change?🔗 🔗
• What is the derivative of a function at a given point? What does this derivative value measure? How do we interpret the derivative value graphically?🔗 🔗
• How are limits used formally in the computation of derivatives?🔗 🔗

🔗

Subsection1.3.1Introduction

The instantaneous rate of change of a function is an idea that sits at the foundation of calculus. It is a generalization of the notion of instantaneous velocity and measures how fast a particular function is changing at a given input. If the original function represents the position of a moving object, this instantaneous rate of change is precisely the instantaneous velocity of the object. In other contexts, instantaneous rate of change could measure the number of cells added to a bacteria culture per day, the number of additional gallons of gasoline consumed per mile by increasing a car’s velocity one mile per hour, or the number of dollars added to a mortgage payment for each percentage point increase in interest rate. The instantaneous rate of change can also be interpreted geometrically on the function’s graph, and this connection is fundamental to many of the main ideas in calculus.🔗 Recall that for a moving object with position function \(s\text{,}\) its average velocity on the time interval \(t = a\) to \(t = a+h\) is given by the quotient \begin{equation*} AV_{[a,a+h]} = \frac{s(a+h)-s(a)}{h}\text{.} \end{equation*} 🔗 In a similar way, we make the following definition for an arbitrary function \(y = f(x)\text{.}\)🔗

Definition1.3.1.

For a function \(f\text{,}\) the average rate of change of \(f\) on the interval \([a,a+h]\) is given by the value \begin{equation*} AV_{[a,a+h]} = \frac{f(a+h)-f(a)}{h}\text{.} \end{equation*} Equivalently, the average rate of change of \(f\) on \([a,b]\) is \begin{equation*} AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}\text{.} \end{equation*} 🔗 🔗It is essential to understand how the average rate of change of \(f\) on an interval is connected to its graph.🔗

  

Preview Activity1.3.1.

Suppose that \(f\) is the function given by the graph below and that \(a\) and \(a+h\) are the input values as labeled on the \(x\)-axis. Use the graph in Figure 1.3.2 to answer the following questions.🔗

The figure shows a portion of the graph of the cubic polynomial \(y = f(x) = 0.1 x (x+2) (x-3) + 2\) in the first quadrant. On the \(x\)-axis, two locations are marked: \(a\) and \(a+h\text{,}\) where both \(a\) and \(h\) are positive numbers. The points that lie on the curve directly above \(a\) and \(a+h\) are identified with small filled circles.🔗

🔗

(a)

Locate and label the points \((a,f(a))\) and \((a+h, f(a+h))\) on the graph.🔗 🔗

(b)

Construct a right triangle whose hypotenuse is the line segment from \((a,f(a))\) to \((a+h,f(a+h))\text{.}\) What are the lengths of the respective legs of this triangle?🔗 🔗

(c)

What is the slope of the line that connects the points \((a,f(a))\) and \((a+h, f(a+h))\text{?}\)🔗 🔗

(d)

Write a meaningful sentence that explains how the average rate of change of the function on a given interval and the slope of a related line are connected.🔗 🔗🔗🔗🔗

Subsection1.3.2The Derivative of a Function at a Point

Just as we defined instantaneous velocity in terms of average velocity, we now define the instantaneous rate of change of a function at a point in terms of the average rate of change of the function \(f\) over related intervals. This instantaneous rate of change of \(f\) at \(a\) is called “the derivative of \(f\) at \(a\text{,}\)” and is denoted by \(f'(a)\text{.}\)🔗

Definition1.3.3.

Let \(f\) be a function and \(x = a\) an input value in the function’s domain. We define the derivative of \(f\) with respect to \(x\) evaluated at \(x = a\), denoted \(f'(a)\text{,}\) by the formula \begin{equation*} f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\text{,} \end{equation*} provided this limit exists. 🔗 🔗Aloud, we read the symbol \(f'(a)\) as either “\(f\)-prime at \(a\)” or “the derivative of \(f\) evaluated at \(x = a\text{.}\)” Much of our work in Chapters 1-3 will be devoted to understanding, computing, applying, and interpreting derivatives. For now, we observe the following important things.🔗

Note1.3.4.

• The derivative of \(f\) at the value \(x = a\) is defined as the limit of the average rate of change of \(f\) on the interval \([a,a+h]\) as \(h \to 0\text{.}\) This limit may not exist, so not every function has a derivative at every point. 🔗
• We say that a function is differentiable at \(x = a\) if it has a derivative at \(x = a\text{.}\) 🔗
• The derivative is a generalization of the instantaneous velocity of a position function: if \(y = s(t)\) is a position function of a moving body, \(s'(a)\) tells us the instantaneous velocity of the body at time \(t=a\text{.}\) 🔗
• Because the units on \(\frac{f(a+h)-f(a)}{h}\) are “units of \(f(x)\) per unit of \(x\text{,}\)” the derivative has these very same units. For instance, if \(s\) measures position in feet and \(t\) measures time in seconds, the units on \(s'(a)\) are feet per second. 🔗
• Because the quantity \(\frac{f(a+h)-f(a)}{h}\) represents the slope of the line through \((a,f(a))\) and \((a+h, f(a+h))\text{,}\) when we compute the derivative we are taking the limit of a collection of slopes of lines. Thus, the derivative itself represents the slope of a particularly important line. 🔗

🔗 🔗We first consider the derivative at a given value as the slope of a certain line.🔗 When we compute an instantaneous rate of change, we allow the interval \([a,a+h]\) to shrink as \(h \to 0\text{.}\) We can think of one endpoint of the interval as “sliding towards” the other. In particular, provided that \(f\) has a derivative at \((a,f(a))\text{,}\) the point \((a+h,f(a+h))\) will approach \((a,f(a))\) as \(h \to 0\text{.}\) Because the process of taking a limit is a dynamic one, it can be helpful to use computing technology to visualize it. One option is an interactive graphic in which the user is able to control the point that is moving. For a helpful collection of examples, consider the work of David Austin of Grand Valley State University, and this particularly relevant example. For interactives that have been built in Geogebra 1 You can even consider building your own examples; the fantastic program Geogebra is available for free download and is easy to learn and use., see Marc Renault’s library via Shippensburg University, with this example being especially fitting for our work in this section.🔗 Figure 1.3.5 shows a sequence of figures with several different lines through the points \((a, f(a))\) and \((a+h,f(a+h))\text{,}\) generated by different values of \(h\text{.}\) These lines (shown in the first three figures in magenta), are often called secant lines to the curve \(y = f(x)\text{.}\) A secant line to a curve is simply a line that passes through two points on the curve. For each such line, the slope of the secant line is \(m = \frac{f(a+h) - f(a)}{h}\text{,}\) where the value of \(h\) depends on the location of the point we choose. We can see in the diagram how, as \(h \to 0\text{,}\) the secant lines start to approach a single line that passes through the point \((a,f(a))\text{.}\) If the limit of the slopes of the secant lines exists, we say that the resulting value is the slope of the tangent line to the curve. This tangent line (shown in the right-most figure in green) to the graph of \(y = f(x)\) at the point \((a,f(a))\) has slope \(m = f'(a)\text{.}\)🔗

 This figure shows four adjacent graphs of the same function (\(y = f(x) = 0.1 x (x+2) (x-3) + 2\) in the first quadrant) and focuses how we think about identifying the tangent line to graph at \((a,f(a))\text{.}\)🔗 The first image shows the points \((a,f(a))\) and \((a+h,f(a+h))\) where \(h\) is a positive number, as well as the secant line that joins these two points.🔗 The second image again shows the point \((a,f(a))\text{,}\) but now the point \((a+h,f(a+h))\) is adjusted to have \(h\) be about half the size of \(h\) in the first image so that \((a+h,f(a+h))\) is closer on the curve to \((a,f(a))\text{.}\) The secant line that joins \((a,f(a))\) and \((a+h,f(a+h))\) is again shown.🔗 The third image is similar to the first and second. The point\((a,f(a))\) remains fixed, but the point \((a+h,f(a+h))\) is updated again to have \(h\) be about half the size of \(h\) in the second image so that \((a+h,f(a+h))\) is even closer on the curve to \((a,f(a))\text{.}\) Here the points \((a,f(a))\) and \((a+h,f(a+h))\) are very close to one another, andthe secant line that joins them is again shown.🔗 The fourth and final image shows the point \((a,f(a))\) and the tangent line to the curvey \(y = f(x)\) at that point. Up close, the tangent line looks indistinguishable from the curve itself.🔗

If the tangent line at \(x = a\) exists, the graph of \(f\) looks like a straight line when viewed up close at \((a,f(a))\text{.}\) In Figure 1.3.6 we combine the four graphs in Figure 1.3.5 into the single one on the left, and zoom in on the box centered at \((a,f(a))\) on the right. Observe how the tangent line sits relative to the curve \(y = f(x)\) at \((a,f(a))\) and how closely it resembles the curve near \(x = a\text{.}\)🔗

 This figure shows two different perspectives on the graphs in Figure 1.3.5.🔗 The first image shows the four plots of three secant lines and the tangent line to \(y=f(x)\) on the same graph. There is also a small box centered on the point \((a,f(a))\text{.}\) In this image, especially within the box, the secant lines appear to get closer to the tangent line as \(h\) gets closer to zero.🔗 The second image zooms in on the contents of the box and focuses on the point \((a,f(a))\text{,}\) the tangent line, and the secant line that results from the point \((a+h,f(a+h))\) that is closest to \((a,f(a))\) from the preceding image. The tangent line looks very similar to the curve \(y=f(x)\) (especially near the point \((a,f(a))\)), and the secant line that is shown looks like a just-slightly-tilted version of the tangent line that is not as closely aligned with \(y=f(x)\text{.}\)🔗

Note1.3.7.

The instantaneous rate of change of \(f\) with respect to \(x\) at \(x = a\text{,}\) \(f'(a)\text{,}\) also measures the slope of the tangent line to the curve \(y = f(x)\) at \((a,f(a))\text{.}\)🔗 🔗The following example demonstrates several key ideas involving the derivative of a function.🔗

Example1.3.8.Using the limit definition of the derivative.

For the function \(f(x) = x - x^2\text{,}\) use the limit definition of the derivative to compute \(f'(2)\text{.}\) In addition, discuss the meaning of this value and draw a labeled graph that supports your explanation.🔗 Solution. From the limit definition, we know that \begin{equation*} f'(2) = \lim_{h \to 0} \frac{f(2+h)-f(2)}{h}\text{.} \end{equation*} 🔗 Now we use the rule for \(f\text{,}\) and observe that \(f(2) = 2 - 2^2 = -2\) and \(f(2+h) = (2+h) - (2+h)^2\text{.}\) Substituting these values into the limit definition, we have that \begin{equation*} f'(2) = \lim_{h \to 0} \frac{(2+h) - (2+h)^2 - (-2)}{h}\text{.} \end{equation*} 🔗 In order to let \(h \to 0\text{,}\) we must simplify the quotient. Expanding and distributing in the numerator, \begin{equation*} f'(2) = \lim_{h \to 0} \frac{2+h - 4 - 4h - h^2 + 2}{h}\text{.} \end{equation*} 🔗 Combining like terms, we have \begin{equation*} f'(2) = \lim_{h \to 0} \frac{ -3h - h^2}{h}\text{.} \end{equation*} 🔗 Next, we remove a common factor of \(h\) in both the numerator and denominator and find that \begin{equation*} f'(2) = \lim_{h \to 0} (-3-h)\text{.} \end{equation*} 🔗 Finally, we are able to take the limit as \(h \to 0\text{,}\) and thus conclude that \(f'(2) = -3\text{.}\) We note that \(f'(2)\) is the instantaneous rate of change of \(f\) at the point \((2,-2)\text{.}\) It is also the slope of the tangent line to the graph of \(y = x - x^2\) at the point \((2,-2)\text{.}\) Figure 1.3.9 shows both the function and the line through \((2,-2)\) with slope \(m = f'(2) = -3\text{.}\)🔗

 This figure shows a graph of the quadratic function \(y = x - x^2\) along with its tangent line at the point \((2,-2)\text{.}\) The values of \(x\) range horizontally from \(-1\) to \(3\text{;}\) values of \(y\) range vertically from \(-6\) to \(1\text{.}\) The grid boxes are \(1 \times 1\text{.}\)🔗 The graph of \(y = x - x^2\) is a parabola that opens down with vertex at \((\frac12, \frac14)\) and \(x\)-intercepts at \((0,0)\) and \((1,0)\text{.}\) The tangent line passes through \((2,-2)\) on the curve and has slope \(m = -3\text{.}\) Near \(x=2\text{,}\) the curve and the tangent line align perfectly and look indistinguishable.🔗

🔗 🔗The following activities will help you explore a variety of key ideas related to derivatives.🔗

  

Activity1.3.2.

Consider the function \(f\) whose formula is \(\displaystyle f(x) = 3 - 2x\text{.}\)🔗 🔗

(a)

What familiar type of function is \(f\text{?}\) What can you say about the slope of \(f\) at every value of \(x\text{?}\)🔗 🔗

(b)

Compute the average rate of change of \(f\) on the intervals \([1,4]\text{,}\) \([3,7]\text{,}\) and \([5,5+h]\text{;}\) simplify each result as much as possible. What do you notice about these quantities?🔗 🔗

(c)

Use the limit definition of the derivative to compute the exact instantaneous rate of change of \(f\) with respect to \(x\) at the value \(a = 1\text{.}\) That is, compute \(f'(1)\) using the limit definition. Show your work. Is your result surprising?🔗 🔗

(d)

Without doing any additional computations, what are the values of \(f'(2)\text{,}\) \(f'(\pi)\text{,}\) and \(f'(-\sqrt{2})\text{?}\) Why?🔗 🔗🔗🔗

  

Activity1.3.3.

A water balloon is tossed vertically in the air from a window. The balloon’s height in feet at time \(t\) in seconds after being launched is given by \(s(t) = -16t^2 + 16t + 32\text{.}\) Use this function to respond to each of the following questions.🔗 🔗

(a)

Sketch an accurate, labeled graph of \(s\) on the axes provided. You should be able to do this without using computing technology.🔗 A coordinate grid for plotting the function and its tangent line. The values of \(t\) range horizontally from \(-0.5\) to \(2.5\text{;}\) values of \(y\) range vertically from \(-8\) to \(40\text{.}\)🔗 🔗

(b)

Compute the average rate of change of \(s\) on the time interval \([1,2]\text{.}\) Include units on your answer and write one sentence to explain the meaning of the value you found.🔗 🔗

(c)

Use the limit definition to compute the instantaneous rate of change of \(s\) with respect to time, \(t\text{,}\) at the instant \(a = 1\text{.}\) Show your work using proper notation, include units on your answer, and write one sentence to explain the meaning of the value you found.🔗 🔗

(d)

On your graph in (a), sketch two lines: one whose slope represents the average rate of change of \(s\) on \([1,2]\text{,}\) the other whose slope represents the instantaneous rate of change of \(s\) at the instant \(a=1\text{.}\) Label each line clearly.🔗 🔗

(e)

For what values of \(a\) do you expect \(s'(a)\) to be positive? Why? Answer the same questions when “positive” is replaced by “negative” and “zero.”🔗 🔗🔗🔗

  

Activity1.3.4.

A rapidly growing city in Arizona has its population \(P\) at time \(t\text{,}\) where \(t\) is the number of decades after the year 2010, modeled by the formula \(P(t) = 25000 e^{t/5}\text{.}\) Use this function to respond to the following questions.🔗 🔗

(a)

Sketch an accurate graph of \(P\) for \(t = 0\) to \(t = 5\) on the axes provided. Label the scale on the axes carefully.🔗 A grid an axes for plotting the function \(P(t)\) in the first quadrant. The grid is \(6\) units wide and tall, with no labeled scale, and the coordinate axes lie one unit up from the bottom and one unit in from the left.🔗 🔗

(b)

Compute the average rate of change of \(P\) between 2030 and 2050. Include units on your answer and write one sentence to explain the meaning (in everyday language) of the value you found.🔗 🔗

(c)

Use the limit definition to write an expression for the instantaneous rate of change of \(P\) with respect to time, \(t\text{,}\) at the instant \(a = 2\text{.}\) Explain why this limit is difficult to evaluate exactly.🔗 🔗

(d)

Estimate the limit in (c) for the instantaneous rate of change of \(P\) at the instant \(a = 2\) by using several small \(h\) values. Once you have determined an accurate estimate of \(P'(2)\text{,}\) include units on your answer, and write one sentence (using everyday language) to explain the meaning of the value you found.🔗 🔗

(e)

On your graph, sketch two lines: one whose slope represents the average rate of change of \(P\) on \([2,4]\text{,}\) the other whose slope represents the instantaneous rate of change of \(P\) at the instant \(a=2\text{.}\)🔗 🔗

(f)

In a carefully-worded sentence, describe the behavior of \(P'(a)\) as \(a\) increases in value. What does this reflect about the behavior of the given function \(P\text{?}\)🔗 🔗🔗🔗🔗

Subsection1.3.3Summary

• The average rate of change of a function \(f\) on the interval \([a,b]\) is \(AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}\text{.}\) The units on the average rate of change are “units of \(f(x)\) per unit of \(x\)”, and the numerical value of the average rate of change represents the slope of the secant line between the points \((a,f(a))\) and \((b,f(b))\) on the graph of \(y = f(x)\text{.}\) If we view the interval as being \([a,a+h]\) instead of \([a,b]\text{,}\) the meaning is still the same, but the average rate of change is now computed by \(AV_{[a,b]} = \frac{f(a+h)-f(a)}{h}\text{.}\) 🔗
• The instantaneous rate of change with respect to \(x\) of a function \(f\) at a value \(x = a\) is denoted \(f'(a)\) (read “the derivative of \(f\) evaluated at \(a\)” or “\(f\)-prime at \(a\)”) and is defined by the formula \begin{equation*} f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\text{,} \end{equation*} provided the limit exists. Note particularly that the instantaneous rate of change at \(x = a\) is the limit of the average rate of change on \([a,a+h]\) as \(h \to 0\text{,}\) and that its units are also “units of \(f(x)\) per unit of \(x\)”. 🔗
• Provided the derivative \(f'(a)\) exists, its value tells us the instantaneous rate of change of \(f\) with respect to \(x\) at \(x = a\text{,}\) which geometrically is the slope of the tangent line to the curve \(y = f(x)\) at the point \((a,f(a))\text{.}\) We even say that \(f'(a)\) is the “slope of the curve” at the point \((a,f(a))\text{.}\) 🔗
• Limits allow us to move from the rate of change over an interval to the rate of change at a single point. 🔗

🔗 🔗

Exercises1.3.4Exercises

1.

Activate Consider the graph of the function \(f(x)\) shown below.🔗 Using this graph, for each of the following pairs of numbers decide which is larger. Be sure that you can explain your answer.🔗 A. \(f(3)\)

• <🔗 🔗
• =🔗 🔗
• >🔗 🔗

\(f(4)\) 🔗 B. \(f(3) - f(2)\)

• <🔗 🔗
• =🔗 🔗
• >🔗 🔗

\(f(2) - f(1)\) 🔗 C. \(\frac{f(2) - f(1)}{2 - 1}\)

• <🔗 🔗
• =🔗 🔗
• >🔗 🔗

\(\frac{f(3) - f(1)}{3 - 1}\) 🔗 D. \(f'(1)\)

• <🔗 🔗
• =🔗 🔗
• >🔗 🔗

\(f'(4)\) 🔗 🔗

2.

Activate Match the points labeled on the curve below with the given slopes in the following table.🔗

slope	-3	-1	-1/2	0	1	2
label	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗	A 🔗 B 🔗 C 🔗 D 🔗 E 🔗 F 🔗

🔗

3.

Activate If \(f(x) = \frac { 2 } {x^2}\text{,}\) find \(f'( 3 )\text{,}\) using the definition of derivative.🔗 \(f'( 3 )\) is the limit as \(h\to\) of the expression🔗 The value of this limit is 🔗 🔗

4.

Activate Let \(f(x)\) be the function whose graph is shown below.🔗 Graph of a piecewise function consisting of a horizontal line on \(0\leq x \leq 3\text{,}\) a nonhorizontal line on \(3\leq x \leq 5\text{,}\) and a parabola on \(5\leq x \leq 9\) with the vertex of the parabola occurring at \(x=7\text{.}\)🔗 Determine \(f'(a)\) for \(a = 1,2,4,7\text{.}\)🔗 \(f'(1) =\) 🔗 \(f'(2) =\) 🔗 \(f'(4) =\) 🔗 \(f'(7) =\) 🔗 🔗

5.

Activate Consider the function \(y = f(x)\) graphed below.🔗 Give the \(x\)-coordinate of a point where:🔗 A. the derivative of the function is negative: \(x =\) 🔗 B. the value of the function is negative: \(x =\) 🔗 C. the derivative of the function is smallest (most negative): \(x =\) 🔗 D. the derivative of the function is zero: \(x =\) 🔗 E. the derivative of the function is approximately the same as the derivative at \(x = 3.25\) (be sure that you give a point that is distinct from \(x = 3.25\text{!}\)): \(x =\) 🔗 🔗

6.

Activate The figure below shows a function \(g(x)\) and its tangent line at the point \(B = (5.4,4.4)\text{.}\) If the point \(A\) on the tangent line is \((5.33,4.38)\text{,}\) fill in the blanks below to complete the statements about the function \(g\) at the point \(B\text{.}\)🔗 Graph of a sharply concave down function with a steep initial slope, bending down as the x-values increase. The point B is on the curve near the initial bend, the tangent line extends through B, and the point A is on the tangent line to the left of B.🔗 \(g(\) \() =\) 🔗 \(g'(\) \() =\) 🔗 🔗

7.

Activate Let \(f(x) = x^2 + 4 x\text{.}\)🔗 (A) Find the slope of the secant line joining \((1, f(1))\) and \((9, f(9))\text{.}\)🔗 Slope of secant line = 🔗 (B) Find the slope of the secant line joining \((4, f(4))\) and \((4 + h, f(4 + h))\text{.}\)🔗 Slope of secant line = 🔗 (C) Find the slope of the tangent line at \((4, f(4))\text{.}\)🔗 Slope of tangent line = 🔗 (D) Find the equation of the tangent line at \((4, f(4))\text{.}\)🔗 \(y =\) 🔗 🔗

8.

Activate Find \(f'(a)\) for \begin{equation*} f(x) = 7+11x-3x^{2}. \end{equation*} \(f'(a)\) = 🔗 🔗

9.

Activate Find \(f'(a)\) for \begin{equation*} f(x) =\frac{9}{\sqrt{7-8 x}}. \end{equation*} 🔗 \(f'(a)\) = 🔗 🔗

10.

Consider the graph of \(y = f(x)\) provided in Figure 1.3.10.🔗

1. On the graph of \(y = f(x)\text{,}\) sketch and label the following quantities:
   • the secant line to \(y = f(x)\) on the interval \([-3,-1]\) and the secant line to \(y = f(x)\) on the interval \([0,2]\text{.}\) 🔗
   • the tangent line to \(y = f(x)\) at \(x = -3\) and the tangent line to \(y = f(x)\) at \(x = 0\text{.}\) 🔗
   🔗 🔗
2. What is the approximate value of the average rate of change of \(f\) on \([-3,-1]\text{?}\) On \([0,2]\text{?}\) How are these values related to your work in (a)? 🔗
3. What is the approximate value of the instantaneous rate of change of \(f\) at \(x = -3\text{?}\) At \(x = 0\text{?}\) How are these values related to your work in (a)? 🔗

🔗

Plot of \(f(x)=0.1 x (x+2) (x-3) + 2\) on the interval \(-5 \lt x \lt 5\text{.}\) The \(y\)-values also are shown on the interval \(-5 \lt y \lt 5\text{.}\)🔗

🔗

11.

For each of the following prompts, sketch a graph on the provided axes in Figure 1.3.11 of a function that has the stated properties.🔗

Two adjacent coordinate axes for plotting the described functions. Each grid has \(x\) ranging horizontally from \(-3.5\) to \(-3.5\text{,}\) and \(y\) ranging vertically from \(-3.5\) to \(-3.5\text{.}\)🔗

1. \(y = f(x)\) such that
   • the average rate of change of \(f\) on \([-3,0]\) is \(-2\) and the average rate of change of \(f\) on \([1,3]\) is 0.5, and 🔗
   • the instantaneous rate of change of \(f\) at \(x = -1\) is \(-1\) and the instantaneous rate of change of \(f\) at \(x = 2\) is 1. 🔗
   🔗 🔗
2. \(y = g(x)\) such that
   • \(\frac{g(3)-g(-2)}{5} = 0\) and \(\frac{g(1)-g(-1)}{2} = -1\text{,}\) and 🔗
   • \(g'(2) = 1\) and \(g'(-1) = 0\) 🔗
   🔗 🔗

🔗 🔗

12.

Suppose that the population, \(P\text{,}\) of China (in billions) can be approximated by the function \(P(t) = 1.15(1.014)^t\) where \(t\) is the number of years since the start of 1993.🔗

1. According to the model, what was the total change in the population of China between January 1, 1993 and January 1, 2000? What will be the average rate of change of the population over this time period? Is this average rate of change greater or less than the instantaneous rate of change of the population on January 1, 2000? Explain and justify, being sure to include proper units on all your answers. 🔗
2. According to the model, what is the average rate of change of the population of China in the ten-year period starting on January 1, 2012? 🔗
3. Write an expression involving limits that, if evaluated, would give the exact instantaneous rate of change of the population on today’s date. Then estimate the value of this limit (discuss how you chose to do so) and explain the meaning (including units) of the value you have found. 🔗
4. Find an equation for the tangent line to the function \(y = P(t)\) at the point where the \(t\)-value is given by today’s date. 🔗

🔗 🔗

13.

The goal of this problem is to compute the value of the derivative at a point for several different functions, where for each one we do so in three different ways, and then to compare the results to see that each produces the same value.🔗 For each of the following functions, use the limit definition of the derivative to compute the value of \(f'(a)\) using three different approaches: strive to use the algebraic approach first (to compute the limit exactly), then test your result using numerical evidence (with small values of \(h\)), and finally plot the graph of \(y = f(x)\) near \((a,f(a))\) along with the appropriate tangent line to estimate the value of \(f'(a)\) visually. Compare your findings among all three approaches; if you are unable to complete the algebraic approach, still work numerically and graphically.🔗

1. \(f(x) = x^2 - 3x\text{,}\) \(a = 2\) 🔗
2. \(f(x) = \frac{1}{x}\text{,}\) \(a = 1\) 🔗
3. \(f(x) = \sqrt{x}\text{,}\) \(a = 1\) 🔗
4. \(f(x) = 2 - |x-1|\text{,}\) \(a = 1\) 🔗
5. \(f(x) = \sin(x)\text{,}\) \(a = \frac{\pi}{2}\) 🔗

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