Chapter 5

Suppose that \(f(a)=g(a)\) and that \(f^{\prime}(x) \leq g^{\prime}(x)\) for all \(x \geq a\). (a) Prove that \(f(x) \leq g(x)\) for all \(x \geq a\). (b) Use part (a) to prove that \(e^{x} \geq 1+x\) for all \(x \geq 0\). (c) Use parts (a) and (b) to prove that \(e^{x} \geq 1+x+\frac{x^{2}}{2}\) for all \(x \geq 0\). (d) Can you generalize these results?

Find the absolute extrema for \(f(x)=-\frac{x+4}{x-4}\) on [0,3] .

A light shines from the top of a pole \(20 \mathrm{~m}\) high. An object is dropped from the same height from a point \(10 \mathrm{~m}\) away, so that its height at time seconds is \(h(t)=20-9.8 t^{2} / 2 .\) How fast is the object's shadow moving on the ground one second later?

A piece of cardboard is 1 meter by \(1 / 2\) meter. A square is to be cut from each corner and the sides folded up to make an open-top box. What are the dimensions of the box with maximum possible volume?

Find all local maximum and minimum points by the second derivative test. $$ y=x^{2}-x $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{x^{2}}{e^{x}-x-1} $$

Find the absolute extrema for \(f(x)=-\frac{x+4}{x-4}\) on [0,3] .

(a) A square piece of cardboard of side a is used to make an open-top box by cutting out a small square from each corner and bending up the sides. How large a square should be cut from each corner in order that the box have maximum volume? \((b)\) What if the piece of cardboard used to make the box is a rectangle of sides a and b?

Find all local maximum and minimum points by the second derivative test. $$ y=2+3 x-x^{3} $$

Compute the following limits. $$ \lim _{x \rightarrow 1} \frac{\ln x}{x-1} $$

Find the absolute extrema for \(f(x)=\ln (x) / x^{2}\) on [1,4] .

A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top; the colored glass transmits only \(1 / 2\) as much light per unit area as the the clear glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and the window may be no more than 1.5 meters wide, find the dimensions of the rectangular portion of the window that lets through the most light.

Find all local maximum and minimum points by the second derivative test. $$ y=x^{3}-9 x^{2}+24 x $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\ln \left(x^{2}+1\right)}{x} $$

Find the absolute extrema for \(f(x)=x \sqrt{1-x^{2}}\) on [-1,1] .

A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only \(k\) times as much light per unit area as the clear glass \((k\) is between 0 and 1\() .\) If the distance from top to bottom (across both the rectangle and the semicircle) is a fixed distance \(H,\) find \((\) in terms of \(k)\) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light.

Find all local maximum and minimum points by the second derivative test. $$ y=x^{4}-2 x^{2}+3 $$

Compute the following limits. $$ \lim _{x \rightarrow 1} \frac{x \ln x}{x^{2}-1} $$

Find the absolute extrema for \(f(x)=x e^{-x^{2} / 32}\) on [0,2] .

You are designing a poster to contain a fixed amount A of printing (measured in square centimeters) and have margins of a centimeters at the top and bottom and b centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amount of posterboard needed.

Find all local maximum and minimum points by the second derivative test. $$ y=3 x^{4}-4 x^{3} $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\sin (2 x)}{\ln (x+1)} $$

Find the absolute extrema for \(f(x)=x-\tan ^{-1}(2 x)\) on [0,2] .

What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?

Find all local maximum and minimum points by the second derivative test. $$ y=\left(x^{2}-1\right) / x $$

Compute the following limits. $$ \lim _{x \rightarrow 1} \frac{x^{1 / 4}-1}{x} $$

The U.S. post office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 in. Find the dimensions of the largest acceptable box with square front and back.

Find all local maximum and minimum points by the second derivative test. $$ y=3 x^{2}-\left(1 / x^{2}\right) $$

Compute the following limits. $$ \lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1} $$

For each of the following, sketch a potential graph of a continuous function on the closed interval [0,4] with the given properties. (a) Absolute minimum at \(0,\) absolute maximum at \(2,\) local minimum at 3 . (b) Absolute maximum at \(1,\) absolute minimum at \(2,\) local maximum at \(3 .\) (c) Absolute minimum at \(4,\) absolute maximum at \(1,\) local minimum at \(2,\) local maxima at 1 and 3 .

Find the dimensions of the lightest cylindrical can containing 0.25 liter \(\left(=250 \mathrm{~cm}^{3}\right)\) if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side.

Find all local maximum and minimum points by the second derivative test. $$ y=\cos (2 x)-x $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{3 x^{2}+x+2}{x-4} $$

A conical paper cup is to hold \(1 / 4\) of a liter. Find the height and radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula \(\pi r \sqrt{r^{2}+h^{2}}\) for the area of the side of a cone.

Find all local maximum and minimum points by the second derivative test. $$ y=4 x+\sqrt{1-x} $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{\sqrt{x+4}-2} $$

A conical paper cup is to hold a fixed volume of water. Find the ratio of height to base radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula \(\pi r \sqrt{r^{2}+h^{2}}\) for the area of the side of a cone, called the lateral area of the cone.

Find all local maximum and minimum points by the second derivative test. $$ y=(x+1) / \sqrt{5 x^{2}+35} $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{\sqrt{x+2}-2} $$

Find the fraction of the area of a triangle that is occupied by the largest rectangle that can be drawn in the triangle (with one of its sides along a side of the triangle). Show that this fraction does not depend on the dimensions of the given triangle.

Find all local maximum and minimum points by the second derivative test. $$ y=x^{5}-x $$

Compute the following limits. $$ \lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+1}+1}{\sqrt{x+1}-1} $$

Find all local maximum and minimum points by the second derivative test. $$ y=6 x+\sin 3 x $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+1}-1}{\sqrt{x+1}-1} $$

Find all local maximum and minimum points by the second derivative test. $$ y=x+1 / x $$

Compute the following limits. $$ \lim _{x \rightarrow 1}(x+5)\left(\frac{1}{2 x}+\frac{1}{x+2}\right) $$

Find all local maximum and minimum points by the second derivative test. $$ y=x^{2}+1 / x $$

Compute the following limits. $$ \lim _{x \rightarrow 2} \frac{x^{3}-6 x-2}{x^{3}+4} $$

Find all local maximum and minimum points by the second derivative test. $$ y=(x+5)^{1 / 4} $$

Discuss what happens if we try to use \(L\) 'Hôpital's rule to find the limit \(\lim _{x \rightarrow \infty} \frac{x+\sin x}{x+1}\).