Chapter 4

Explain how the first derivative of a function determines where the function is increasing and decreasing.

Fill in the blanks with either of the words the derivative or an antiderivative: If \(F^{\prime}(x)=f(x),\) then \(f\) is _______of \(F\) and \(F\) is ____________ of \(f\).

Explain with examples what is meant by the indeterminate form \(0 / 0\)

Sketch the graph of a smooth function \(f\) and label a point \(P(a,(f(a))\) on the curve. Draw the line that represents the linear approximation to \(f\) at \(P\).

Fill in the blanks: The goal of an optimization problem is to find the maximum or minimum value of the _________ function subject _______ to the ____.

Why is it important to determine the domain of \(f\) before graphing \(f ?\)

Explain how the iteration formula for Newton's method works.

Describe the set of antiderivatives of \(f(x)=0\)

Why are special methods, such as l'Hôpital's Rule, needed to evaluate indeterminate forms (as opposed to substitution)?

Suppose you find the linear approximation to a differentiable function at a local maximum of that function. Describe the graph of the linear approximation.

Why is it important to determine the domain of \(f\) before graphing \(f ?\)

Sketch the graph of a function that has neither a local maximum nor a local minimum at a point where \(f^{\prime}(x)=0\)

Describe the set of antiderivatives of \(f(x)=1\)

Explain the steps used to apply l'Hôpital's Rule to a limit of the form \(0 / 0\)

Explain why Rolle's Theorem cannot be applied to the function \(f(x)=|x|\) on the interval \([-a, a],\) for any \(a>0\)

How is linear approximation used to approximate the value of a function \(f\) near a point at which \(f\) and \(f^{\prime}\) are easily evaluated?

Suppose the objective function is \(Q=x^{2} y\) and you know that \(x+y=10 .\) Write the objective function first in terms of \(x\) and then in terms of \(y\)

Can the graph of a polynomial have vertical or horizontal asymptotes? Explain.

What conditions must be met to ensure that a function has an absolute maximum value and an absolute minimum value on an interval?

Explain how to apply the Second Derivative Test.

Give the formula for Newton's method for the function \(f(x)=x^{2}-5\).

Why do two different antiderivatives of a function differ by a constant?

To which indeterminate forms does 1 'Hópital's Rule apply directly?

How can linear approximation be used to approximate the change in \(y=f(x)\) given a change in \(x ?\)

Suppose you wish to minimize a continuous objective function on a closed interval, but you find that it has only a single local maximum. Where should you look for the solution to the problem?

Where are the vertical asymptotes of a rational function located?

Suppose \(f^{\prime \prime}\) exists and is positive on an interval \(I\). Describe the relationship between the graph of \(f\) and its tangent lines on the interval \(I\)

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{2}-6 ; x_{0}=3$$

Give the antiderivatives of \(x^{p}\). For what values of \(p\) does your answer apply?

Given a function \(f\) that is differentiable on its domain, write and explain the relationship between the differentials \(d x\) and \(d y\).

Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)

How do you find the absolute maximum and minimum values of a function that is continuous on a closed interval?

Sketch the graph of a function that has an absolute maximum, a local minimum, but no absolute minimum on [0,3].

Sketch a function that changes from concave up to concave down as \(x\) increases. Describe how the second derivative of this function changes.

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{2}-2 x-3 ; x_{0}=2$$

Give the antideriyatives of \(e^{-x}\)

Give an example of a limit of the form \(\infty / \infty\) as \(x \rightarrow 0\).

At what points \(c\) does the conclusion of the Mean Value Theorem hold for \(f(x)=x^{3}\) on the interval [-10,10]\(?\)

Of all rectangles with a fixed perimeter of \(P,\) which one has the maximum area? (Give the dimensions in terms of \(P .\) )

Describe the possible end behavior of a polynomial.

What is an inflection point?

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=e^{-x}-x ; x_{0}=\ln 2$$

Give the antiderivatives of \(1 / x\)

Explain why the form \(1^{\infty}\) is indeterminate and cannot be evaluated by substitution. Explain how the competing functions behave.

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. $$f(x)=x(x-1)^{2} ;[0,1]$$

Sketch a curve with the following properties. $$\begin{aligned} &f^{\prime}<0 \text { and } f^{\prime \prime}<0, \text { for } x<3\\\ &f^{\prime}<0 \text { and } f^{\prime \prime}>0, \text { for } x>3 \end{aligned}$$

Sketch the graph of a function \(f\) that has a local maximum value at a point \(c\) where \(f^{\prime}(c)=0\).

Give a function that does not have an inflection point at a point where \(f^{\prime \prime}(x)=0\)

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{3}-2 ; x_{0}=2$$

Evaluate \(\int \cos a x d x\) and \(\int \sin a x d x,\) where \(a\) is a constant.