Chapter 5

Find the dimensions of the rectangle of largest area having fixed perimeter \(100 .\)

Compute the following limits. \(\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin x}\)

Find the linearization \(L(x)\) of \(f(x)=\ln (1+x)\) at \(a=0 .\) Use this linearization to approximate \(f(0.1)\)

Let \(f(x)=x^{2} .\) Find a value \(c \in(-1,2)\) so that \(f^{\prime}(c)\) equals the slope between the endpoints of \(f(x)\) on [-1,2]

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=x^{2}-x $$

Air is being pumped into a spherical balloon at a constant rate of \(3 \mathrm{~cm}^{3} / \mathrm{s} .\) How fast is the radius of the balloon increasing when the radius reaches \(5 \mathrm{~cm} ?\)

Find the dimensions of the rectangle of largest area having fixed perimeter \(P\).

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=2+3 x-x^{3} $$

Compute the following limits. $$ \lim _{x \rightarrow \infty} \frac{e^{x}}{x^{3}} $$

Verify that \(f(x)=x /(x+2)\) satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and then find all of the values, \(c,\) that satisfy the conclusion of the theorem.

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=2+3 x-x^{3} $$

A cylindrical tank standing upright (with one circular base on the ground) has radius 20 \(\mathrm{cm} .\) How fast does the water level in the tank drop when the water is being drained at \(25 \mathrm{~cm}^{3} / \mathrm{sec} ?\)

A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base.

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=x^{3}-9 x^{2}+24 x $$

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

Show in detail that the linear approximation of \(\sin x\) at \(x=0\) is \(L(x)=x\) and the linear approximation of \(\cos x\) at \(x=0\) is \(L(x)=1\).

Verify that \(f(x)=3 x /(x+7)\) satisfies the hypotheses of the Mean Value Theorem on the interval [-2,6] and then find all of the values, \(c,\) that satisfy the conclusion of the theorem.

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=x^{3}-9 x^{2}+24 x $$

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=x^{3}-9 x^{2}+24 x $$

Compute the following limits. $$ \lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}} $$

Use \(f(x)=\sqrt[3]{x+1}\) to approximate \(\sqrt[3]{9}\) by choosing an appropriate point \(x=a\). Are we over- or under-estimating the value of \(\sqrt[3]{9}\) ? Explain.

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

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=x^{4}-2 x^{2}+3 $$

A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of \(0.6 \mathrm{~m} / \mathrm{sec} .\) How fast is the top sliding down the wall when the foot of the ladder is \(5 \mathrm{~m}\) from the wall?

A box with square base and no top is to hold a volume V. Find (in terms of V) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve V.)

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=3 x^{4}-4 x^{3} $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{\sqrt{9+x}-3}{x} $$

Let \(f(x)=x^{4}\). If \(a=1\) and \(d x=\Delta x=1 / 2,\) what are \(\Delta y\) and \(d y\) ?

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

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=3 x^{4}-4 x^{3} $$

A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at 0.1 meters per second. How fast is the foot of the ladder approaching the wall when the foot of the ladder is \(5 \mathrm{~m}\) from the wall?

You have 100 feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area?

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

Let \(f(x)=\sqrt{x}\). If \(a=1\) and \(d x=\Delta x=1 / 10,\) what are \(\Delta y\) and \(d y ?\)

Describe all functions with derivative \(x^{2}+47 x-5 .\)

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=\left(x^{2}-1\right) / x $$

A rotating beacon is located 2 miles out in the water. Let \(A\) be the point on the shore that is closest to the beacon. As the beacon rotates at 10 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point 2 miles along the shore from the point \(A ?\)

You have l feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area?

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=3 x^{2}-\left(1 / x^{2}\right) $$

Compute the following limits. $$ \lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{\sqrt[3]{x}-1} $$

Let \(f(x)=\sin (2 x) .\) If \(a=\pi\) and \(d x=\Delta x=\pi / 100,\) what are \(\Delta y\) and \(d y ?\)

Describe all functions with derivative \(\frac{1}{1+x^{2}} .\)

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ y=3 x^{2}-\left(1 / x^{2}\right) $$

A baseball diamond is a square \(90 \mathrm{ft}\) on a side. A player runs from first base to second base at \(15 \mathrm{ft} / \mathrm{sec} .\) At what rate is the player's distance from third base decreasing when she is half way from first to second base?

Marketing tells you that if you set the price of an item at $$\$ 10$$ then you will be unable to sell it, but that you can sell 500 items for each dollar below $$\$10$$ that you set the price. Suppose your fixed costs total $$\$ 3000,$$ and your marginal cost is $$\$ 2$$$ per item. What is the most profit you can make?

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ y=\cos (2 x)-x $$

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{(1-x)^{1 / 4}-1}{x} $$

Use differentials to estimate the amount of paint needed to apply a coat of paint \(0.02 \mathrm{~cm}\) thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius \(r\) is \(V=(4 / 3) \pi r^{3}\). Notice that you are given that \(d r=0.02 .)\)

Describe all functions with derivative \(x^{3}-\frac{1}{x}\)

Sand is poured onto a surface at \(15 \mathrm{~cm}^{3} / \mathrm{sec}\), forming a conical pile whose base diameter is always equal to its altitude. How fast is the altitude of the pile increasing when the pile is \(3 \mathrm{~cm}\) high?