Chapter 8

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ f(x)=\frac{x}{x+1} $$

In Exercises, find \(d y / d x\) implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating \(d y / d x\) at the point. $$ 4 y^{2}-x^{2}=7 $$

In Exercises, find the higher-order derivative. $$ f^{\prime \prime}(x)=2 x^{2}+7 x-12 $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ g(x)=2 x^{4}-8 x^{3}+12 x^{2}+12 x $$

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ y=\left\\{\begin{array}{ll} 4-x^{2}, & x \leq 0 \\ -2 x, & x>0 \end{array}\right. $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=x^{3}-9 x^{2}+27 x-27 $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(x)=-4 x^{3}-8 x^{2}+32 $$

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=4 \sqrt{x}-2 x+1, \quad[0,6] $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ y=\left\\{\begin{array}{ll} 2 x+1, & x \leq-1 \\ x^{2}-2, & x>-1 \end{array}\right. $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=3 x^{3}-9 x+1 $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ h(x)=(x-2)^{3}(x-1) $$

In Exercises, find the absolute extrema of the function on the interval \([0, \infty)\). $$ f(x)=\frac{4 x}{x^{2}+1} $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ y=\left\\{\begin{array}{ll} 3 x+1, & x \leq 1 \\ 5-x^{2}, & x>1 \end{array}\right. $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=(x+3)(x-4)(x+5) $$

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(t)=(1-t)(t-4)\left(t^{2}-4\right) $$

In Exercises, find the absolute extrema of the function on the interval \([0, \infty)\). $$ f(x)=\frac{8}{x+1} $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. $$ y=\left\\{\begin{array}{ll} -x^{3}+1, & x \leq 0 \\ -x^{2}+2 x, & x>0 \end{array}\right. $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=(x+2)(x-2)(x+3)(x-3) $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=x^{3}-12 x $$

In Exercises, find the absolute extrema of the function on the interval \([0, \infty)\). $$ f(x)=\frac{2 x}{x^{2}+4} $$

The ordering and transportation cost \(C\) (in hundreds of dollars) for an automobile dealership is modeled by \(C=10\left(\frac{1}{x}+\frac{x}{x+3}\right), x \geq 1\) where \(x\) is the number of automobiles ordered. (a) Find the intervals on which \(C\) is increasing or decreasing. (b) Use a graphing utility to graph the cost function. (c) Use the trace feature to determine the order sizes for which the cost is \(\$ 900\). Assuming that the revenue function is increasing for \(x \geq 0\), which order size would you use? Explain your reasoning.

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=x \sqrt{x^{2}-1} $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=x^{3}-3 x $$

In Exercises, find the absolute extrema of the function on the interval \([0, \infty)\). $$ f(x)=8-\frac{4 x}{x^{2}+1} $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=x \sqrt{4-x^{2}} $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=x^{3}-6 x^{2}+12 x $$

The number \(y\) of medical degrees conferred in the United States from 1970 through 2004 can be modeled by \(y=0.813 t^{3}-55.70 t^{2}+1185.2 t+7752, \quad 0 \leq t \leq 34\) where \(t\) is the time in years, with \(t=0\) corresponding to 1970. (a) Use a graphing utility to graph the model. Then graphically estimate the years during which the model is increasing and the years during which it is decreasing. (b) Use the test for increasing and decreasing functions to verify the result of part (a).

Demand In Exercises, find the rate of change of \(x\) with respect to \(p\). $$ p=\frac{2}{0.00001 x^{3}+0.1 x} \quad x \geq 0 $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=\frac{x}{x^{2}+3} $$

In Exercises, find the maximum value of \(\left|f^{\prime \prime}(x)\right|\) on the closed interval. (You will use this skill in Section \(12.4\) to estimate the error in the Trapezoidal Rule.) $$ f(x)=\sqrt{1+x^{3}}, \quad[0,2] $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=x^{3}-\frac{3}{2} x^{2}-6 x $$

The profit \(P\) made by a cinema from selling \(x\) bags of popcorn can be modeled by \(P=2.36 x-\frac{x^{2}}{25,000}-3500, \quad 0 \leq x \leq 50,000\) (a) Find the intervals on which \(P\) is increasing and decreasing. (b) If you owned the cinema, what price would you charge to obtain a maximum profit for popcorn? Explain your reasoning.

Demand In Exercises, find the rate of change of \(x\) with respect to \(p\). $$ p=\frac{4}{0.000001 x^{2}+0.05 x+1} \quad x \geq 0 $$

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=\frac{x}{x-1} $$

In Exercises, find the maximum value of \(\left|f^{\prime \prime}(x)\right|\) on the closed interval. (You will use this skill in Section \(12.4\) to estimate the error in the Trapezoidal Rule.) $$ f(x)=\frac{1}{x^{2}+1}, \quad[0,3] $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=\frac{1}{4} x^{4}-2 x^{2} $$

A fast-food restaurant determines the cost and revenue models for its hamburgers. \(C=0.6 x+7500, \quad 0 \leq x \leq 50,000\) \(R=\frac{1}{20,000}\left(65,000 x-x^{2}\right), \quad 0 \leq x \leq 50,000\) (a) Write the profit function for this situation. (b) Determine the intervals on which the profit function is increasing and decreasing. (c) Determine how many hamburgers the restaurant needs to sell to obtain a maximum profit. Explain your reasoning.

Demand In Exercises, find the rate of change of \(x\) with respect to \(p\). $$ p=\sqrt{\frac{200-x}{2 x}}, 0

A ball is propelled straight upward from ground level with an initial velocity of 144 feet per second. (a) Write the position, velocity, and acceleration functions of the ball. (b) When is the ball at its highest point? How high is this point? (c) How fast is the ball traveling when it hits the ground? How is this speed related to the initial velocity?

In Exercises, find the maximum value of \(\left|f^{(4)}(x)\right|\) on the closed interval. (You will use this skill in Section \(12.4\) to estimate the error in Simpson's Rule.) $$ f(x)=(x+1)^{2 / 3}, \quad[0,2] $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ f(x)=2 x^{4}-8 x+3 $$

Demand In Exercises, find the rate of change of \(x\) with respect to \(p\). $$ p=\sqrt{\frac{500-x}{2 x}}, 0

A brick becomes dislodged from the top of the Empire State Building (at a height of 1250 feet) and falls to the sidewalk below. (a) Write the position, velocity, and acceleration functions of the brick. (b) How long does it take the brick to hit the sidewalk? (c) How fast is the brick traveling when it hits the sidewalk?

In Exercises, find the maximum value of \(\left|f^{(4)}(x)\right|\) on the closed interval. (You will use this skill in Section \(12.4\) to estimate the error in Simpson's Rule.) $$ f(x)=\frac{1}{x^{2}+1}, \quad[-1,1] $$

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=(x-2)(x+1)^{2} $$

In Exercises, graph a function on the interval \([-2,5]\) having the given characteristics. Absolute maximum at \(x=-2\) Absolute minimum at \(x=1\) Relative maximum at \(x=3\)

The velocity (in feet per second) of an automobile starting from rest is modeled by \(\frac{d s}{d t}=\frac{90 t}{t+10}\) Create a table showing the velocity and acceleration at 10 -second intervals during the first minute of travel. What can you conclude?

In Exercises, use a graphing utility to graph the function and identify all relative extrema and points of inflection. $$ g(x)=(x-6)(x+2)^{3} $$

In Exercises, graph a function on the interval \([-2,5]\) having the given characteristics. Relative minimum at \(x=-1\) Critical number at \(x=0\), but no extrema Absolute maximum at \(x=2\) Absolute minimum at \(x=5\)