Chapter 4

In Exercises \(53-56,\) find a value \(c\) whose existence is guaranteed by the Mean Value Theorem applied to the given function \(f\) on the interval \(I=[a, b]\). $$ f(x)=x+1 / x, \quad I=[1,2] $$

Plot the function $$ f(x)=\frac{3 x^{5}+10000 x^{4}}{x^{5}+10000 x^{4}+x+10000} $$ over the interval \([-15,15] .\) What appears to be the asymptote for \(f ?\) Analyze \(f\) for asymptotes and plot \(f\) in a window that better illustrates its asymptote.

An arrow is shot at a \(60^{\circ}\) angle to the horizontal with initial velocity \(190 \mathrm{ft} / \mathrm{s}\). How high will the arrow travel? What will be the horizontal component of its velocity at height 50 feet (going up)?

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\sinh (x)+2 \cosh (x) $$

Calculate the given limit. \(\lim _{x \rightarrow 0} \frac{\cosh (x)-1}{\sinh ^{2}(x)}\)

Plot \(f(x)=12 x^{5}-2565 x^{4}+146200 x^{3}+1\) for \(x \in I=\) 0,300]\(.\) Plot \(f^{\prime}\) and \(f^{\prime \prime}\) for \(x \in I\). Is \(f\) increasing on \(I ?\) Is \(f^{\prime}>0\) on \(I\) ? Is the graph of \(f\) concave up on \(I\) ? Is \(f^{\prime \prime}>0\) on \(I\) ?

In a light rail system, the Transit Authority schedules a 10 mile trip between two stops to be made in 20 minutes. The train accelerates and decelerates at the rate of \(1 / 3 \mathrm{mi} / \mathrm{min}^{2}\) If the engineer spends the same amount of time accelerating and decelerating, what is the top speed during the trip?

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\tanh (x)-\exp (2 x) $$

Calculate the given limit. \(\lim _{x \rightarrow \infty} \frac{\tanh (x)}{\arctan (x)}\)

The engineer of a freight train needs to stop in 1700 feet in order not to strike a barrier. The train is traveling at a speed of \(30 \mathrm{mi} / \mathrm{hr}\). The engineer applies one set of brakes, which causes him to decelerate at the rate of \(1 / 5 \mathrm{mi} / \mathrm{min}^{2}\) Fifteen seconds later, realizing that he is not going to make it, he applies a second set of brakes, which, together with the first set, causes the train to decelerate at a rate of \(3 / 10\) \(\mathrm{mi} / \mathrm{min}^{2}\). Will the train strike the barrier? If not, with how many feet to spare? If so, at what speed is the impact?

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=2 \sinh ^{-1}(x)-\ln (x) $$

Calculate the given limit. \(\lim _{x \rightarrow 0} \frac{\arcsin (x)-x}{\arctan (x)-x}\)

Suppose that a uniform rod of length \(\ell\) and mass \(m\) can rotate freely about one end. If a point mass \(6 m\) is attached to the rod a distance \(x\) from the pivot, then the period of small oscillations is equal to $$ 2 \pi \sqrt{\frac{2}{3 g}} \cdot \sqrt{\frac{\ell^{2}+18 x^{2}}{12 x+\ell}} $$ For what value of \(x\) is the period least?

A speeding car passes a policeman who is equipped with a radar gun. The policeman determines that the car is doing \(85 \mathrm{mi} / \mathrm{hr}\). By the time the policeman is ready to give chase, the car has a 15 second lead. The policeman has been trained to catch vehicles within 2 minutes of the beginning of pursuit. At what constant rate does the police car need to be able to accelerate to catch the speeding car?

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\tanh ^{-1}(\sqrt{x})-2 \sqrt{x} $$

Calculate the given limit. \(\lim _{x \rightarrow 0} \frac{\operatorname{arcsinh}(x)-\arcsin (x)}{x^{3}}\)

A volume \(V_{0}\) of gas is held at pressure \(p_{0}\) in a reservoir. The gas is discharged through a nozzle of opening area \(A\) into a region at lower pressure \(p\). Then the rate of discharge (in units of weight/time) is given by $$ A \sqrt{\frac{2 g \gamma}{\gamma-1} \frac{p_{0}}{V_{0}}\left(\left(\frac{p}{p_{0}}\right)^{2 / \gamma}-\left(\frac{p}{p_{0}}\right)^{(\gamma+1) / \gamma}\right)} $$ where \(\gamma\) is the adiabatic constant of the gas. What is \(p / p_{0}\) when the rate of discharge is greatest?

In each of Exercises \(57-60,\) a continuous function \(f\) is given on an interval \(I=[a, b] .\) Sketch the graph of \(y=f(x)\) for \(x\) in \(I\). In each case, explain why there can be no \(c\) in \((a, b)\) for which equation (4.2.1) holds. Explain why the Mean Value Theorem is not contradicted. $$ f(x)=\left\\{\begin{array}{cl} x-5 & \text { if }-3 \leq x \leq-1 \\ -x-7 & \text { if }-1

Let \(f\) be differentiable on, \(\mathbb{R}\). Suppose that \(f^{\prime}(2)>0\). Is \(f(2.000001)>f(2) ?\) Explain your answer.

The formula $$ \frac{\left|f^{\prime \prime}(x)\right|}{\left(1+\left(f^{\prime}(x)\right)^{2}\right)^{3 / 2}} $$ measures the curvature of the graph of \(y=f(x)\) at the point $(x, f(x)) . Compute the curvature of the given function. $$ f(x)=\cos (x) $$

In Exercises \(61-64\), use the logarithm to reduce the indeterminate form \(1^{\infty}\) to one that can be handled with l'Hôpital's Rule. \(\lim _{x \rightarrow 0}(\cos (2 x))^{1 / x^{2}}\)

Visitors to the top of the Empire State Building in New York City are cautioned not to throw objects off the roof. How much damage could a one ounce weight dropped from the roof do? The height of the building is 1250 feet. Using the quantity mass \(\times\) velocity, which is momentum, as a measure of the impact of the weight when it strikes, compare your answer with the impact of a ten ounce hammer being swung at a speed of ten feet per second.

True or false: If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a polynomial and if \(f\) has two local minima, then it has a local maximum. Explain your answer.

The formula $$ \frac{\left|f^{\prime \prime}(x)\right|}{\left(1+\left(f^{\prime}(x)\right)^{2}\right)^{3 / 2}} $$ measures the curvature of the graph of \(y=f(x)\) at the point $(x, f(x)) . Compute the curvature of the given function. $$ f(x)=\sqrt{x} $$

Use the logarithm to reduce the indeterminate form \(1^{\infty}\) to one that can be handled with l'Hôpital's Rule. \(\lim _{x \rightarrow \infty}(\sqrt{1-1 / x})^{x}\)

A model rocket is launched straight up with an initial velocity of \(100 \mathrm{ft} / \mathrm{s} .\) The fuel on board the rocket lasts for 8 seconds and maintains the rocket at this upward velocity (countering the negative acceleration due to gravity). After the fuel is spent then gravity takes over. What is the greatest height that the rocket reaches? With what velocity does the rocket strike the ground?

Prove that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable, and \(f^{\prime}>0\) everywhere, then \(f\) is one-to-one. Prove that if \(f^{\prime}<0\) everywhere, then \(f\) is one-to-one.

The formula $$ \frac{\left|f^{\prime \prime}(x)\right|}{\left(1+\left(f^{\prime}(x)\right)^{2}\right)^{3 / 2}} $$ measures the curvature of the graph of \(y=f(x)\) at the point $(x, f(x)) . Compute the curvature of the given function. $$ f(x)=1 / x $$

Use the logarithm to reduce the indeterminate form \(1^{\infty}\) to one that can be handled with l'Hôpital's Rule. \(\lim _{x \rightarrow \infty}\left(\frac{5+x}{2+x}\right)^{x}\)

Find the absolute minimum value and absolute maximum value of $$ f(x)=\frac{x^{2}+\sin (x)}{\sqrt{x^{4}+2 x+2}}, \quad-4 \leq x \leq 4. $$

Show that \(3 x^{4}-4 x^{3}+6 x^{2}-12 x+5=0\) has at most two real valued solutions.

The initial temperature \(T(0)\) of an object is \(50^{\circ} \mathrm{C}\). If the object is cooling at the rate $$T^{\prime}(t)=-0.2 e^{-0.02 t}$$ when measured in degrees centigrade per second, then what is the limit of its temperature as \(t\) tends to infinity?

If \(f\) is increasing on an interval \(I,\) does it follow that \(f^{2}\) is increasing? What if the range of \(f\) is \((0, \infty)\) ?

Use the logarithm to reduce the indeterminate form \(1^{\infty}\) to one that can be handled with l'Hôpital's Rule. \(\lim _{x \rightarrow 0}(1+\sin (x))^{\csc (x)}\)

Suppose that the cost function for producing a certain product is \(C(x)=100000+1000 \sqrt{x}+x^{2} / 40\). Find the production level \(x_{0}\) at which average \(\operatorname{cost} \bar{C}(x)=C(x) / x\) is minimized. What is \(\bar{C}\left(x_{0}\right)\) ? What is the marginal price at this production level?

Show that \(x^{3}-3 x^{2}+4 x-1=0\) has exactly one real root.

At a given moment in time, two race cars are abreast and traveling at the speed of \(90 \mathrm{mi} / \mathrm{hr}\). Car A then begins to accelerate at a constant rate of \(6 \mathrm{mi} / \mathrm{min}^{2}\) and Car \(\mathrm{B}\) begins to accelerate at a constant rate of \(9 \mathrm{mi} / \mathrm{min}^{2}\). The cars are driving on a track of circumference 2 miles. How long will it take Car \(\mathrm{B}\) to lap Car A three times?

Use l'Hôpital's Rule to check that \(\lim _{h \rightarrow 0^{+}}(1+h)^{1 / h}=e\). Now calculate \(\lim _{h \rightarrow 0^{+}}(1+h x)^{1 / h}\) for any fixed \(x\)

Two heat sources separated by a distance \(10 \mathrm{~cm}\) are located on the \(x\) -axis. The heat received by any point on the \(x\) -axis from each of these sources is inversely proportional to the square of its distance from the source. Suppose that the heat received a distance \(1 \mathrm{~cm}\) from one source is twice that received \(1 \mathrm{~cm}\) from the other source. What is the location of the coolest point between the two sources?

If a car that is initially moving at \(100 \mathrm{~km} / \mathrm{hr}\) decelerates to 0 at a constant rate \(r\) in \(60 \mathrm{~m}\), what is \(r ?\)

Suppose that a real-valued function \(f\) is defined and increasing on an interval \((a, b)\). Let \(J\) be the image of \(f\). Without using calculus, prove that \(f^{-1}: J \rightarrow(a, b)\) exists and is increasing. Now assume that \(f^{\prime}\) exists and is positive on the interval \((a, b)\). Give a calculus proof that \(f^{-1}\) is increasing.

Use l'Hôpital's Rule to calculate $$ \lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}} $$ for a function \(f\) that is twice continuously differentiable at \(x\)

Suppose that, for a beam of length 11 meters, the deflection from the horizontal of a point a distance \(x\) from one end is proportional to $$ 2 x^{4}-33 x^{3}+1331 x $$ Determine the point at which the deflection is greatest.

Use Rolle's Theorem to show that \(p(x)=x^{3}+a x^{2}+b\) cannot have three negative roots

If the air resistance to a falling object is proportional to the velocity of that object, then the velocity of that object when dropped from a height \(H\) is $$v(t)=-k g\left(1-e^{-t / k}\right)$$ Here \(k\) is a positive constant and \(g\) is the constant acceleration due to gravity. Calculate the height \(y\) of the object as a function of \(t\).

Without using calculus, prove that the composition of two increasing functions is increasing. Now assume that \(f\) and \(g\) are differentiable functions with positive derivatives and that \(g \circ f\) is defined. Use calculus to show that \(g \circ f\) is increasing.

Let \(a\) and \(b\) be constants with \(b \neq 1 .\) Use l'Hôpital's Rule to calculate $$ \lim _{x \rightarrow 0} \frac{a \sin (x)-\sin (a x)}{\tan (b x)-b \tan (x)} $$

Supposes that a shotputter releases the shot at height \(h\) with angle of inclination \(\alpha,\) and initial speed \(v .\) Then the horizontal distance \(R\) that the shot travels is given by $$ R=\frac{v^{2} \sin (2 \alpha)+v \sqrt{v^{2} \sin ^{2}(2 \alpha)+8 g h \cos ^{2}(\alpha)}}{2 g} $$ Use a computer algebra system to find the value of \(\alpha\) that maximizes \(R\).

Use Rolle's Theorem to prove that \(p(x)=x^{3}+a x^{2}+b\) cannot have two distinct negative roots if \(a<0 .\)

Show that the curvature of the circle \(x^{2}+y^{2}=r^{2}\) is \(1 / r\) at all points \((x, y)\) with \(y \neq 0\).