What's The Graph Look Like? - Ximera

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Mathematical Expression Editor

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Main Part

Ximera tutorial

How to use Ximera

This course is built in Ximera.

How is my work scored?

We explain how your work is scored.

Understanding functions

Same or different?

Two young mathematicians examine one (or two!) functions.

For each input, exactly one output

We define the concept of a function.

Compositions of functions

We discuss compositions of functions.

Inverses of functions

Here we “undo” functions.

Review of famous functions

How crazy could it be?

Two young mathematicians think about the plots of functions.

Polynomial functions

Polynomials are some of our favorite functions.

Rational functions

Rational functions are functions defined by fractions of polynomials.

Trigonometric functions

We review trigonometric functions.

Exponential and logarithmic functions

Exponential and logarithmic functions illuminated.

What is a limit?

Stars and functions

Two young mathematicians discuss stars and functions.

What is a limit?

We introduce limits.

Continuity

The limit of a continuous function at a point is equal to the value of the function at that point.

Limit laws

Equal or not?

Here we see a dialogue where students discuss combining limits with arithmetic.

The limit laws

We give basic laws for working with limits.

The Squeeze Theorem

The Squeeze theorem allows us to compute the limit of a difficult function by “squeezing" it between two easy functions.

(In)determinate forms

Could it be anything?

Two young mathematicians investigate the arithmetic of large and small numbers.

Limits of the form zero over zero

We want to evaluate limits for which the Limit Laws do not apply.

Limits of the form nonzero over zero

What can be said about limits that have the form nonzero over zero?

Using limits to detect asymptotes

Zoom out

Two young mathematicians discuss what curves look like when one “zooms out.”

Vertical asymptotes

We explore functions that “shoot to infinity” near certain points.

Horizontal asymptotes

We explore functions that behave like horizontal lines as the input grows without bound.

Continuity and the Intermediate Value Theorem

Roxy and Yuri like food

Two young mathematicians discuss the eating habits of their cats.

Continuity of piecewise functions

Here we use limits to check whether piecewise functions are continuous.

The Intermediate Value Theorem

Here we see a consequence of a function being continuous.

An application of limits

Limits and velocity

Two young mathematicians discuss limits and instantaneous velocity.

Instantaneous velocity

We use limits to compute instantaneous velocity.

Definition of the derivative

Slope of a curve

Two young mathematicians discuss the novel idea of the “slope of a curve.”

The definition of the derivative

We compute the instantaneous growth rate by computing the limit of average growth rates.

Derivatives as functions

Wait for the right moment

Two young mathematicians discuss derivatives as functions.

The derivative as a function

Here we study the derivative of a function, as a function, in its own right.

Differentiability implies continuity

We see that if a function is differentiable at a point, then it must be continuous at that point.

Rules of differentiation

Patterns in derivatives

Two young mathematicians think about “short cuts” for differentiation.

Basic rules of differentiation

We derive the constant rule, power rule, and sum rule.

The derivative of the natural exponential function

We derive the derivative of the natural exponential function.

The derivative of sine

We derive the derivative of sine.

Product rule and quotient rule

Derivatives of products are tricky

Two young mathematicians discuss derivatives of products and products of derivatives.

The Product rule and quotient rule

Here we compute derivatives of products and quotients of functions

Chain rule

An unnoticed composition

Two young mathematicians discuss the chain rule.

The chain rule

Here we compute derivatives of compositions of functions

Derivatives of trigonometric functions

We use the chain rule to unleash the derivatives of the trigonometric functions.

Higher order derivatives and graphs

Rates of rates

Two young mathematicians look at graph of a function, its first derivative, and its second derivative.

Higher order derivatives and graphs

Here we make a connection between a graph of a function and its derivative and higher order derivatives.

Concavity

Here we examine what the second derivative tells us about the geometry of functions.

Position, velocity, and acceleration

Here we discuss how position, velocity, and acceleration relate to higher derivatives.

Implicit differentiation

Standard form

Two young mathematicians discuss the standard form of a line.

Implicit differentiation

In this section we differentiate equations that contain more than one variable on one side.

Derivatives of inverse exponential functions

We derive the derivatives of inverse exponential functions using implicit differentiation.

Logarithmic differentiation

Multiplication to addition

Two young mathematicians think about derivatives and logarithms.

Logarithmic differentiation

We use logarithms to help us differentiate.

Derivatives of inverse functions

We can figure it out

Two young mathematicians discuss the derivative of inverse functions.

Derivatives of inverse trigonometric functions

We derive the derivatives of inverse trigonometric functions using implicit differentiation.

The Inverse Function Theorem

We see the theoretical underpinning of finding the derivative of an inverse function at a point.

More than one rate

A changing circle

Two young mathematicians discuss a circle that is changing.

More than one rate

Here we work abstract related rates problems.

Applied related rates

Pizza and calculus, so cheesy

Two young mathematicians discuss tossing pizza dough.

Applied related rates

We solve related rates problems in context.

Maximums and minimums

More coffee

Two young mathematicians witness the perils of drinking too much coffee.

Maximums and minimums

We use derivatives to help locate extrema.

Concepts of graphing functions

What’s the graph look like?

Two young mathematicians discuss how to sketch the graphs of functions.

Concepts of graphing functions

We use the language of calculus to describe graphs of functions.

Computations for graphing functions

Wanted: graphing procedure

Two young mathematicians discuss how to sketch the graphs of functions.

Computations for graphing functions

We will give some general guidelines for sketching the plot of a function.

Mean Value Theorem

Let’s run to class

Two young mathematicians race to math class.

The Extreme Value Theorem

We examine a fact about continuous functions.

The Mean Value Theorem

Here we see a key theorem of calculus.

Linear approximation

Replacing curves with lines

Two young mathematicians discuss linear approximation.

Linear approximation

We use a method called “linear approximation” to estimate the value of a (complicated) function at a given point.

Explanation of the product and chain rules

We give explanation for the product rule and chain rule.

Optimization

A mysterious formula

Two young mathematicians discuss optimization from an abstract point of view.

Basic optimization

Now we put our optimization skills to work.

Applied optimization

Volumes of aluminum cans

Two young mathematicians discuss optimizing aluminum cans.

Applied optimization

Now we put our optimization skills to work.

L’Hopital’s rule

A limitless dialogue

Two young mathematicians consider a way to compute limits using derivatives.

L’Hopital’s rule

We use derivatives to give us a “short-cut” for computing limits.

Antiderivatives

Jeopardy! Of calculus

If the answer is a derivative, we seek the questions that give that answer.

Basic antiderivatives

We introduce antiderivatives.

Falling objects

We study a special type of differential equation.

Approximating the area under a curve

What is area?

Two young mathematicians discuss the idea of area.

Introduction to sigma notation

We introduce sigma notation.

Approximating area with rectangles

We introduce the basic idea of using rectangles to approximate the area under a curve.

Definite integrals

Computing areas

Two young mathematicians discuss cutting up areas.

The definite integral

Definite integrals compute net area.

Antiderivatives and area

Meaning of multiplication

A dialogue where students discuss multiplication.

Relating velocity, displacement, antiderivatives and areas

We give an alternative interpretation of the definite integral and make a connection between areas and antiderivatives.

First Fundamental Theorem of Calculus

What’s in a calculus problem?

Two young mathematicians discuss what calculus is all about.

The First Fundamental Theorem of Calculus

The rate that accumulated area under a curve grows is described identically by that curve.

Second Fundamental Theorem of Calculus

A secret of the definite integral

Two young mathematicians discuss what calculus is all about.

The Second Fundamental Theorem of Calculus

The accumulation of a rate is given by the change in the amount.

A tale of three integrals

At this point we have three “different” integrals.

The idea of substitution

Geometry and substitution

Two students consider substitution geometrically.

The idea of substitution

We learn a new technique, called substitution, to help us solve problems involving integration.

Working with substitution

Integrals are puzzles!

Two young mathematicians discuss how tricky integrals are puzzles.

Working with substitution

We explore more difficult problems involving substitution.

The Work-Energy Theorem

Substitution is given a physical meaning.

Applications of integrals

What could it represent?

Two young mathematicians discuss whether integrals are defined properly.

Applications of integrals

We give more contexts to understand integrals.

1. mooculus
2. Calculus 1
3. Concepts of graphing functions
4. What’s the graph look like?

Two young mathematicians discuss how to sketch the graphs of functions.

Check out this dialogue between two calculus students (based on a true story):

Devyn Riley, I’ve been thinking about the derivative. Riley It’s all about change. It’s some “change-detector” tool for math. Devyn I know! What’s crazy is that you can use it as a tool for sniffing out dirt on functions. Riley First \(f'\) tells us increasing or decreasing. Devyn Then \(f''\) tells us concavity. Riley From just that we know all local maxes and mins. Devyn And if we use limits, we can find any asymptotes! Riley You know, I’d like to make up a procedure based on all these facts, that would tell me what the graph of any function would look like. Devyn Me too! Let’s get to work! On some interval, we know that \(f'(x)\) is positive and \(f''(x)\) is positive. Which of the following is the best option for the shape of the graph on that interval? On some interval, we know that \(f'(x)\) is negative and \(f''(x)\) is positive. Which of the following is the best option for the shape of the graph on that interval?

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