Chapter 4

Explain the difference between an absolute minimum and a local minimum.

Find the most general antiderivative of the function. (Check your answer by differentiation.) $$f(x)=\frac{1}{2}+\frac{3}{4} x^{2}-\frac{4}{5} x^{3}$$

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always \(23 .\) On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=2 x^{3}+3 x^{2}-36 x\)

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. \(f(x)=5-12 x+3 x^{2}, \quad[1,3]\)

Suppose \(f\) is a continuous function defined on a closed interval \([a, b] .\) (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for \(f ?\) (b) What steps would you take to find those maximum and minimum values?

Find the most general antiderivative of the function. (Check your answer by differentiation.) $$f(x)=8 x^{9}-3 x^{6}+12 x^{3}$$

Find two numbers whose difference is 100 and whose product is a minimum.

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=4 x^{3}+3 x^{2}-6 x+1\)

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. \(f(x)=x^{3}-x^{2}-6 x+2,[0,3]\)

Find the most general antiderivative of the function. (Check your answer by differentiation.) $$f(x)=7 x^{2 / 5}+8 x^{-4 / 5}$$

Suppose the tangent line to the curve \(y=f(x)\) at the point \((2,5)\) has the equation \(y=9-2 x .\) If Newton's method is used to locate a root of the equation \(f(x)=0\) and the initial approximation is \(x_{1}=2,\) find the second approximation \(x_{2}\) .

Find two positive numbers whose product is 100 and whose sum is a minimum.

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{4}-2 x^{2}+3\)

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. \(f(x)=\sqrt{x}-\frac{1}{3} x, \quad[0,9]\)

Find the most general antiderivative of the function. (Check your answer by differentiation.) $$f(x)=\sqrt[3]{x^{2}}+x \sqrt{x}$$

The sum of two positive numbers is \(16 .\) What is the smallest possible value of the sum of their squares?

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=\frac{x}{x^{2}+1}\)

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. \(f(x)=\cos 2 x, \quad[\pi / 8,7 \pi / 8]\)

Find the most general antiderivative of the function (Check your answer by differentiation.) $$f(x)=3 \sqrt{x}-2 \sqrt[3]{x}$$

What is the maximum vertical distance between the line \(y=x+2\) and the parabola \(y=x^{2}\) for \(-1 \leq x \leq 2 ?\)

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=\sin x+\cos x, \quad 0 \leqslant x \leqslant 2 \pi\)

Let \(f(x)=1-x^{2 / 3}\) . Show that \(f(-1)=f(1)\) but there is no number \(c\) in \((-1,1)\) such that \(f^{\prime}(c)=0 .\) Why does this not contradict Rolle's Theorem?

Find the most general antiderivative of the function (Check your answer by differentiation $$f(t)=\frac{3 t^{4}-t^{3}+6 t^{2}}{t^{4}}$$

Use Newton's method with the specified initial approximation \(x_{1}\) to find \(x_{3},\) the third approximation to the root of the given equation. (Give your answer to four decimal places.) \(\frac{1}{3} x^{3}+\frac{1}{2} x^{2}+3=0, \quad x_{1}=-3\)

What is the minimum vertical distance between the parabolas \(y=x^{2}+1\) and \(y=x-x^{2} ?\)

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=\cos ^{2} x-2 \sin x, \quad 0 \leqslant x \leqslant 2 \pi\)

\(7-10=\) Sketch the graph of a function \(f\) that is continuous on \([1,5]\) and has the given properties. Absolute minimum at \(2,\) absolute maximum at 3 local minimum at 4

Find the most general antiderivative of the function (Check your answer by differentiation $$g(t)=\frac{1+t+t^{2}}{\sqrt{t}}$$

Use Newton's method with the specified initial approximation \(x_{1}\) to find \(x_{3},\) the third approximation to the root of the given equation. (Give your answer to four decimal places.) \(x^{5}-x-1=0, \quad x_{1}=1\)

Find the dimensions of a rectangle with perimeter 100 \(\mathrm{m}\) whose area is as large as possible.

Use the guidelines of this section to sketch the curve. $$y=\frac{1}{5} x^{5}-\frac{8}{3} x^{3}+16 x$$

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=e^{2 x}+e^{-x}\)

\(7-10=\) Sketch the graph of a function \(f\) that is continuous on \([1,5]\) and has the given properties. Absolute minimum at \(1,\) absolute maximum at \(5,\) local maximum at \(2,\) local minimum at 4

Find the most general antiderivative of the function (Check your answer by differentiation $$r(\theta)=\sec \theta \tan \theta-2 e^{\theta}$$

Use Newton's method with the specified initial approximation \(x_{1}\) to find \(x_{3},\) the third approximation to the root of the given equation. (Give your answer to four decimal places.) \(x^{7}+4=0, \quad x_{1}=-1\)

Find the dimensions of a rectangle with area 1000 \(\mathrm{m}^{2}\) whose perimeter is as small as possible.

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{2} \ln x\)

\(7-10=\) Sketch the graph of a function \(f\) that is continuous on \([1,5]\) and has the given properties. Absolute maximum at \(5,\) absolute minimum at \(2,\) local maximum at \(3,\) local minima at 2 and 4

Find the most general antiderivative of the function (Check your answer by differentiation $$h(\theta)=2 \sin \theta-\sec ^{2} \theta$$

Consider the following problem: A farmer with 750 \(\mathrm{ft}\) of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw scveral diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, cstimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (f) Finish solving the problem and compare the answer with your estimate in part (a).

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{2}-x-\ln x\)

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. \(f(x)=2 x^{2}-3 x+1,[0,2]\)

Find the most general antiderivative of the function (Check your answer by differentiation $$f(t)=\sin t+2 \sinh t$$

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

\(7-10=\) Sketch the graph of a function \(f\) that is continuous on \([1,5]\) and has the given properties. \(f\) has no local maximum or minimum, but 2 and 4 are critical numbers

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f .\) (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{4} e^{-x}\)

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. \(f(x)=x^{3}-3 x+2,[-2,2]\)

(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2 . (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2 . (c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2 .

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=5 e^{x}-3 \cosh x$$