Chapter 5

Find the area of the largest rectangle that fits inside a semicircle of radius 10 (one side of the rectangle is along the diameter of the semicircle).

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ f(x)=(5-x) /(x+2) $$

Compute the following limits. $$ \lim _{t \rightarrow 0}\left(t+\frac{1}{t}\right)\left((4-t)^{3 / 2}-8\right) $$

Find the \(5^{\text {th }}\) degree Taylor polynomial for \(f(x)=\sin x\) around \(a=0\). (a) Use this Taylor polynomial to approximate \(\sin (0.1)\). (b) Use a calculator to find \(\sin (0.1) .\) How does this compare to our approximation in part \((a) ?\)

Describe all functions with derivative \(\sin (2 x) .\)

Find all local maximum and minimum points \((x, y)\) by the method of this section. $$ f(x)=x^{2}-98 x+4 $$

A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point \(5 \mathrm{ft}\) higher than the front of the boat. The rope is being pulled through the ring at the rate of \(0.6 \mathrm{ft} / \mathrm{sec} .\) How fast is the boat approaching the dock when \(13 \mathrm{ft}\) of rope are out?

Find the area of the largest rectangle that fits inside a semicircle of radius \(r\) ( one side of the rectangle is along the diameter of the semicircle).

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ f(x)=\left|x^{2}-121\right| $$

Compute the following limits. $$ \lim _{t \rightarrow 0^{+}}\left(\frac{1}{t}+\frac{1}{\sqrt{t}}\right)(\sqrt{t+1}-1) $$

Suppose that \(f^{\prime \prime}\) exists and is continuous on \([1,2] .\) Suppose also that \(\left|f^{\prime \prime}(x)\right| \leq \frac{1}{4}\) for all \(x\) in \((1,2) .\) Prove that if we use the linearization \(y=L(x)\) of \(y=f(x)\) at \(x=1\) as an approximation of \(y=f(x)\) near \(x=1,\) then our estimated value of \(f(1.2)\) is guaranteed to have an accuracy of at least \(0.01,\) i.e., our estimate will lie within 0.01 units of the true value.

Find \(f(x)\) if \(f^{\prime}(x)=e^{-x}\) and \(f(0)=2\).

For any real number x there is a unique integer n such that \(n \leq x

A balloon is at a height of 50 meters, and is rising at the constant rate of \(5 \mathrm{~m} / \mathrm{sec} . \mathrm{A}\) bicyclist passes beneath it, traveling in a straight line at the constant speed of \(10 \mathrm{~m} / \mathrm{sec} .\) How fast is the distance between the bicyclist and the balloon increasing 2 seconds later?

For a cylinder with surface area \(50,\) including the top and the bottom, find the ratio of height to base radius that maximizes the volume.

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ f(x)=x^{3} /(x+1) $$

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

Find the \(3^{\text {rd }}\) degree Taylor polynomial for \(f(x)=\frac{1}{1-x}-1\) around \(a=0 .\) Explain why this approximation would not be useful for calculating \(f(5)\).

Suppose that \(f\) is a differentiable function such that \(f^{\prime}(x) \geq-3\) for all \(x\). What is the smallest possible value of \(f(4)\) if \(f(-1)=2 ?\)

Explain why the function \(f(x)=1 / x\) has no local maxima or minima.

A pyramid-shaped vat has square cross-section and stands on its tip. The dimensions at the top are \(2 \mathrm{~m} \times 2 \mathrm{~m},\) and the depth is \(5 \mathrm{~m} .\) If water is flowing into the vat at \(3 \mathrm{~m}^{3} / \mathrm{min},\) how fast is the water level rising when the depth of water (at the deepest point) is \(4 \mathrm{~m} ?\) Note: the volume of any "conical" shape (including pyramids) is (1/3)(height) ( area of base).

For a cylinder with given surface area \(S,\) including the top and the bottom, find the ratio of height to base radius that maximizes the volume.

Find all critical points and identify them as local maximum points, local minimum points, or neither. $$ f(x)=\sin ^{2} x $$

Compute the following limits. $$ \lim _{u \rightarrow 1} \frac{(u-1)^{3}}{(1 / u)-u^{2}+3 / u-3} $$

Consider \(f(x)=\ln x\) around \(a=1 .\) (a) Find a general formula for \(f^{(n)}(x)\) for \(n \geq 1\). (b) Find a general formula for the Taylor Polynomial, \(T_{n}(x)\).

Show that the equation \(6 x^{4}-7 x+1=0\) does not have more than two distinct real roots.

How many critical points can a quadratic polynomial function have?

A woman \(5 \mathrm{ft}\) tall walks at the rate of \(3.5 \mathrm{ft} /\) sec away from a streetlight that is \(12 \mathrm{ft}\) above the ground. \(A t\) what rate is the tip of her shadow moving? At what rate is her shadow lengthening?

You want to make cylindrical containers to hold I liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side \(2 r,\) so that \(2(2 r)^{2}=8 r^{2}\) of material is needed ( rather than \(2 \pi r^{2},\) which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material, and also find the ratio of height to radius for this container.

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{2+(1 / x)}{3-(2 / x)} $$

Use Newton's Method to find all roots of \(f(x)=3 x^{2}-9 x-11\). (Hint: use Intermediate Value Theorem to choose an appropriate \(x_{0}\) )

Let \(f\) be differentiable on \(\mathbb{R}\). Suppose that \(f^{\prime}(x) \neq 0\) for every \(x\). Prove that \(f\) has at most one real root.

Show that a cubic polynomial can have at most two critical points. Give examples to show that a cubic polynomial can have zero, one, or two critical points.

A man 1.8 meters tall walks at the rate of 1 meter per second toward a streetlight that is 4 meters above the ground. \(A t\) what rate is the tip of his shadow moving? \(A t\) what rate is his shadow shortening?

You want to make cylindrical containers of a given volume \(V\) using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side \(2 r,\) so that \(2(2 r)^{2}=8 r^{2}\) of material is needed (rather than \(2 \pi r^{2},\) which is the total area of the top and bottom). Find the optimal ratio of height to radius.

Let \(f(\theta)=\cos ^{2}(\theta)-2 \sin (\theta) .\) Find the intervals where \(f\) is increasing and the intervals where \(f\) is decreasing in \([0,2 \pi] .\) Use this information to classify the critical points of \(f\) as either local maximums, local minimums, or neither.

Compute the following limits. $$ \lim _{x \rightarrow 0^{+}} \frac{1+5 / \sqrt{x}}{2+1 / \sqrt{x}} $$

Prove that for all real \(x\) and \(y|\cos x-\cos y| \leq|x-y| .\) State and prove an analogous result involving sine.

Explore the family of functions \(f(x)=x^{3}+c x+1\) where \(c\) is a constant. How many and what types of local extremes are there? Your answer should depend on the value of \(c,\) that is, different values of \(c\) will give different answers.

A police helicopter is flying at 150 mph at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at \(190 \mathrm{mph} .\) Find the speed of the car:

Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let \(H\) and \(R\) be the height and base radius of the larger cone, and let \(h\) and \(r\) be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating h and r.)

Let \(r>0 .\) Find the local maxima and minima of the function \(f(x)=\sqrt{r^{2}-x^{2}}\) on its domain \([-r, r] .\)

Compute the following limits. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{(\pi / 2)-x} $$

Consider \(f(x)=\sin x .\) What happens when we choose \(x_{0}=\pi / 2 ?\) Explain.

Show that \(\sqrt{1+x} \leq 1+(x / 2)\) if \(-1

We generalize the preceding two questions. Let \(n\) be a positive integer and let \(f\) be a polynomial of degree n. How many critical points can \(f\) have? (Hint: Recall the Fundamental Theorem of Algebra, which says that a polynomial of degree \(n\) has at most \(n\) roots. \()\)

A police helicopter is flying at 200 kilometers per hour at a constant altitude of \(1 \mathrm{~km}\) above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at \(250 \mathrm{kph.}\) Find the speed of the car.

A container holding a fixed volume is being made in the shape of a cylinder with a hemispherical top. (The hemispherical top has the same radius as the cylinder.) Find the ratio of height to radius of the cylinder which minimizes the cost of the container if \((\) a) the cost per unit area of the top is twice as great as the cost per unit area of the side, and the container is made with no bottom; \((b)\) the same as in \((a),\) except that the container is made with a circular bottom, for which the cost per unit area is 1.5 times the cost per unit area of the side.

Let \(f(x)=a x^{2}+b x+c\) with \(a \neq 0\). Show that \(f\) has exactly one critical point using the first derivative test. Give conditions on a and b which guarantee that the critical point will be a maximum. It is possible to see this without using calculus at all; explain.

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