Chapter 4

If 1200 \(\mathrm{cm}^{2}\) of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Use Newton's method to approximate the given number correct to eight decimal places. \(\sqrt[5]{20}\)

Find the local maximum and minimum values of \(f\) using both the First and Second Derivative Tests. Which method do you prefer? \(f(x)=1+3 x^{2}-2 x^{3}\)

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. \(f(x)=\ln x, \quad[1,4]\)

(a) Sketch the graph of a function on \([-1,2]\) that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on \([-1,2]\) that has a local maximum but no absolute maximum.

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=2 \sqrt{x}+6 \cos x$$

A box with a square base and open top must have a volume of \(32,000 \mathrm{cm}^{3} .\) Find the dimensions of the box that minimize the amount of material used.

Find the local maximum and minimum values of \(f\) using both the First and Second Derivative Tests. Which method do you prefer? \(f(x)=\frac{x^{2}}{x-1}\)

Use Newton's method to approximate the given number correct to eight decimal places. \(\sqrt[100]{100}\)

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. \(f(x)=1 / x, \quad[1,3]\)

(a) Sketch the graph of a function on \([-1,2]\) that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on \([-1,2]\) that is discontinuous but has both an absolute maximum and an absolute minimum.

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

(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.

Use Newton's method to approximate the indicated root of the equation correct to six decimal places. The root of \(x^{4}-2 x^{3}+5 x^{2}-6=0\) in the interval \([1,2]\)

Suppose \(f^{\prime \prime}\) is continuous on \((-\infty, \infty)\) (a) If \(f^{\prime}(2)=0\) and \(f^{\prime \prime}(2)=-5,\) what can you say about \(f ?\) (b) If \(f^{\prime}(6)=0\) and \(f^{\prime \prime}(6)=0,\) what can you say about \(f ?\)

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at \((c, f(c))\) . Are the secant line and the tangent line parallel? \(f(x)=\sqrt{x}, \quad[0,4]\)

(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

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

A rectangular storage container with an open top is to have a volume of 10 \(\mathrm{m}^{3} .\) The length of its base is twice the width. Material for the base costs \(\$ 10\) per square meter. Material for the sides costs \(\$ 6\) per square meter. Find the cost of materials for the cheapest such container.

(a) Find the critical numbers of \(f(x)=x^{4}(x-1)^{3}\) (b) What does the Second Derivative Test tell you about the behavior of \(f\) at these critical numbers? (c) What does the First Derivative Test tell you?

Use Newton's method to approximate the indicated root of the equation correct to six decimal places. The positive root of \(3 \sin x=x\)

Find the number \(c\) that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at \((c, f(c))\) . Are the secant line and the tangent line parallel? \(f(x)=e^{-x}, \quad[0,2]\)

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$$

Find the antiderivative \(F\) of \(f\) that satisfies the given condition. Check your answer by comparing the graphs of \(f\) and \(F .\) $$f(x)=5 x^{4}-2 x^{5}, \quad F(0)=4$$

Find the point on the line \(y=2 x+3\) that is closest to the origin.

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(x^{6}-x^{5}-6 x^{4}-x^{2}+x+10=0\)

Let \(f(x)=(x-3)^{-2}\) . Show that there is no value of \(c\) in \((1,4)\) such that \(f(4)-f(1)=f^{\prime}(c)(4-1) .\) Why does this not contradict the Mean Value Theorem?

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$$

Find the antiderivative \(F\) of \(f\) that satisfies the given condition. Check your answer by comparing the graphs of \(f\) and \(F .\) $$f(x)=4-3\left(1+x^{2}\right)^{-1}, \quad F(1)=0$$

Find the point on the curve \(y=\sqrt{x}\) that is closest to the point \((3,0) .\)

Let \(f(x)=2-|2 x-1| .\) Show that there is no value of \(c\) such that \(f(3)-f(0)=f^{\prime}(c)(3-0) .\) Why does this not contradict the Mean Value Theorem?

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\sin x, \quad 0 \leqslant x<\pi / 2$$

Find the points on the ellipse \(4 x^{2}+y^{2}=4\) that are farthest away from the point \((1,0)\) .

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(e^{-x}=2+x\)

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) are always negative.

Show that the equation has exactly one real root. \(2 x+\cos x=0\)

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

Find, correct to two decimal places, the coordinates of the point on the curve \(y=\sin x\) that is closest to the point \((4,2)\)

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(x^{2}\left(4-x^{2}\right)=\frac{4}{x^{2}+1}\)

Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote \(x=0, \quad f^{\prime}(x)>0\) if \(x<-2,\) \(f^{\prime}(x)<0\) if \(x>-2(x \neq 0),\) \(f^{\prime \prime}(x)<0\) if \(x<0, \quad f^{\prime \prime}(x)>0\) if \(x>0\)

Show that the equation has exactly one real root. \(x^{3}+e^{x}=0\)

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=\ln x, \quad 0

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side \(L\) if one side of the rectangle lies on the base of the triangle.

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(x^{2} \sqrt{2-x-x^{2}}=1\)

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0,\) \(f^{\prime}(x)>0\) if \(x < 0\) or \(2 < x < 4,\) \(f^{\prime}(x) < 0\) if \(0 < x < 2\) or \(x>4,\) \(f^{\prime \prime}(x)>0\) if \(1 < x <3, \quad f^{\prime \prime}(x) <0\) if \(x<1\) or \(x>3\)

Show that the equation \(x^{3}-15 x+c=0\) has at most one root in the interval \([-2,2] .\)

Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.

Sketch the graph of a function that satisfies all of the given conditions. \(f^{\prime}(1)=f^{\prime}(-1)=0, \quad f^{\prime}(x)<0\) if \(|x|<1\) \(f^{\prime}(x)>0\) if \(1<|x|<2, \quad f^{\prime}(x)=-1\) if \(|x|>2\) \(f^{\prime \prime}(x)<0\) if \(-2 < x<0, \quad\) inflection point \((0,1)\)

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=1 / x, \quad 1 < x < 3$$

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. \(\cos \left(x^{2}-x\right)=x^{4}\)