Chapter 4

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\ln x}\right)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\sqrt[3]{x^{3}+1}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{4+\sqrt{t}}{t^{3}} d t$$

Find the function with the given derivative whose graph passes through the point \(P\). $$r^{\prime}(t)=\sec t \tan t-1, \quad P(0,0)$$

Projectile motion The range \(R\) of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity \(v_{0}\) at an angle \(\alpha\) with the horizontal, then in Chapter 12 we find that $$R=\frac{v_{0}^{2}}{g} \sin 2 \alpha$$ where \(g\) is the downward acceleration due to gravity. Find the angle \(\alpha\) for which the range \(R\) is the largest possible.

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$f(x)=e^{\sqrt{x}}$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0^{+}}(\csc x-\cot x+\cos x)$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{8 x}{x^{2}+4}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(-2 \cos t) d t$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=9.8 t+5, \quad s(0)=10$$

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$f(x)=x \ln x$$

Find the function's absolute maximum and minimum values and say where they are assumed. $$h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(-5 \sin t) d t$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=32 t-2, \quad s(0.5)=4$$

Stiffness of a beam The stiffness \(S\) of a rectangular beam is proportional to its width times the cube of its depth. a. Find the dimensions of the stiffest beam that can be cut from a 12 -in.-diameter cylindrical log. b. Graph \(S\) as a function of the beam's width \(w,\) assuming the proportionality constant to be \(k=1 .\) Reconcile what you see with your answer in part (a). c. On the same screen, graph \(S\) as a function of the beam's depth \(d,\) again taking \(k=1 .\) Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of \(k ?\) Try it.

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$f(x)=x^{2} \ln x$$

Use I'Hópital's rule to find the limits. $$\lim _{h \rightarrow 0} \frac{e^{h}-(1+h)}{h^{2}}$$

Determine all critical points for each function. $$y=x^{2}-6 x+7$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\left|x^{2}-1\right|$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int 7 \sin \frac{\theta}{3} d \theta$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=\sin \pi t, \quad s(0)=0$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(x)=2 x-x^{2}, \quad-\infty

Frictionless cart \(\quad\) A small frictionless cart, attached to the wall by a spring, is pulled \(10 \mathrm{cm}\) from its rest position and released at time \(t=0\) to roll back and forth for 4 sec. Its position at time \(t\) is \(s=10 \cos \pi t\) a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then? GRAPH CANT COPY

Use I'Hópital's rule to find the limits. $$\lim _{t \rightarrow \infty} \frac{e^{t}+t^{2}}{t^{2}-t}$$

Determine all critical points for each function. $$f(x)=6 x^{2}-x^{3}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int 3 \cos 5 \theta d \theta$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=\frac{2}{\pi} \cos \frac{2 t}{\pi}, \quad s\left(\pi^{2}\right)=1$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(x)=(x+1)^{2}, \quad-\infty

Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t,\) respectively. a. At what times in the interval \(0

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} x^{2} e^{-x}$$

Determine all critical points for each function. $$f(x)=x(4-x)^{3}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\sqrt{|x|}=\left\\{\begin{array}{ll}\sqrt{-x}, & x<0 \\\\\sqrt{x}, & x \geq 0\end{array}\right.$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(-3 \csc ^{2} x\right) d x$$

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity. and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=e^{t}, \quad v(0)=20, \quad s(0)=5$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty$$

Distance between two ships At noon, ship \(A\) was 12 nautical miles due north of ship \(B\). Ship \(A\) was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship \(B\) was sailing east at 8 knots and continued to do so all day. a. Start counting time with \(t=0\) at noon and express the distance \(s\) between the ships as a function of \(t\) b. How rapidly was the distance between the ships changing at noon? One hour later? c. The visibility that day was 5 nautical miles. Did the ships ever sight each other? d. Graph \(s\) and \(d s / d t\) together as functions of \(t\) for \(-1 \leq t \leq 3\) using different colors if possible. Compare the graphs and reconcile what you see with your answers in parts (b) and (c). e. The graph of \(d s / d t\) looks as if it might have a horizontal asymptote in the first quadrant. This in turn suggests that ds/dt approaches a limiting value as \(t \rightarrow \infty .\) What is this value? What is its relation to the ships' individual speeds?

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{x-\sin x}{x \tan x}$$

Determine all critical points for each function. $$g(x)=(x-1)^{2}(x-3)^{2}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\sqrt{|x-4|}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(-\frac{\sec ^{2} x}{3}\right) d x$$

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity. and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=9.8, \quad v(0)=-3, \quad s(0)=0$$

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$g(x)=-x^{2}-6 x-9,-4 \leq x<\infty$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\left(e^{x}-1\right)^{2}}{x \sin x}$$

Determine all critical points for each function. $$y=x^{2}+\frac{2}{x}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x e^{1 / x}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{\csc \theta \cot \theta}{2} d \theta$$

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity. and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=-4 \sin 2 t, \quad v(0)=2, \quad s(0)=-3$$

Tin pest When metallic tin is kept below \(13.2^{\circ} \mathrm{C},\) it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious, and indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases, it is reasonable to assume that the rate \(v=d x / d t\) of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, \(v\) may be considered to be a function of \(x\) alone, and $$v=k x(a-x)=k a x-k x^{2}$$ where \(x=\) the amount of product \(a=\) the amount of substance at the beginning \(k=\) a positive constant. At what value of \(x\) does the rate \(v\) have a maximum? What is the maximum value of \(v ?\)