Chapter 8

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=x^{2}-x-2 $$

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=-2 x^{2}+4 x+3 $$

In Exercises, evaluate the derivative of the function at the indicated points on the graph. $$ f(x)=\frac{x^{2}}{x^{2}+4} $$

In Exercises, use the given values to find \(d y / d t\) and \(d x / d t\) $$ \begin{aligned} &y=\sqrt{x} \quad \text { (a) } \frac{d y}{d t} \quad x=4, \frac{d x}{d t}=3\\\ &\text { (b) } \frac{d x}{d t} \quad x=25, \frac{d y}{d t}=2 \end{aligned} $$

In Exercises, find \(d y / d x\) $$ x y=4 $$

In Exercises, find the second derivative of the function. $$ f(x)=9-2 x $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=-x^{3}+3 x^{2}-2 $$

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=x^{2}+8 x+10 $$

In Exercises, evaluate the derivative of the function at the indicated points on the graph. $$ f(x)=x+\frac{32}{x^{2}} $$

In Exercises, use the given values to find \(d y / d t\) and \(d x / d t\) $$ \begin{aligned} &y=2\left(x^{2}-3 x\right)\\\ &\text { (a) } \frac{d y}{d t} \quad x=3, \frac{d x}{d t}=2\\\ &\text { (b) } \frac{d x}{d t} \quad x=1, \frac{d y}{d t}=5 \end{aligned} $$

In Exercises, find \(d y / d x\) $$ 3 x^{2}-y=8 x $$

In Exercises, find the second derivative of the function. $$ f(x)=4 x+15 $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ f(x)=\frac{x^{2}-1}{2 x+1} $$

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=x^{2}-6 x $$

In Exercises, use the given values to find \(d y / d t\) and \(d x / d t\) \(x y=4\) (a) \(\frac{d y}{d t} \quad x=8, \frac{d x}{d t}=10\) (b) \(\frac{d x}{d t}\) \(x=1, \frac{d y}{d t}=-6\)

In Exercises, find \(d y / d x\) $$ y^{2}=1-x^{2}, 0 \leq x \leq 1 $$

In Exercises, find the second derivative of the function. $$ f(x)=x^{2}+7 x-4 $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ f(x)=\frac{x^{2}+4}{4-x^{2}} $$

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=-4 x^{2}+4 x+1 $$

In Exercises, use the given values to find \(d y / d t\) and \(d x / d t\) \(x^{2}+y^{2}=25\) (a) \(\frac{d y}{d t} \quad x=3, y=4, \frac{d x}{d t}=8\) (b) \(\frac{d x}{d t} \quad x=4, y=3, \frac{d y}{d t}=-2\)

In Exercises, find \(d y / d x\) $$ 4 x^{2} y-\frac{3}{y}=0 $$

In Exercises, find the second derivative of the function. $$ f(x)=3 x^{2}+4 x $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ f(x)=\frac{24}{x^{2}+12} $$

In Exercises, find all relative extrema of the function. $$ g(x)=6 x^{3}-15 x^{2}+12 x $$

In Exercises, use the derivative to identify the open intervals on which the function is increasing or decreasing. Verify your result with the graph of the function. $$ f(x)=-(x+1)^{2} $$

The radius \(r\) of a circle is increasing at a rate of 3 inches per minute. Find the rates of change of the area when (a) \(r=6\) inches and (b) \(r=24\) inches.

In Exercises, find \(d y / d x\) $$ x^{2} y^{2}-2 x=3 $$

In Exercises, find the second derivative of the function. $$ g(t)=\frac{1}{3} t^{3}-4 t^{2}+2 t $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

In Exercises, find all relative extrema of the function. $$ g(x)=\frac{1}{5} x^{5}-x $$

In Exercises, use the derivative to identify the open intervals on which the function is increasing or decreasing. Verify your result with the graph of the function. $$ f(x)=\frac{x^{3}}{4}-3 x $$

The radius \(r\) of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the volume when (a) \(r=6\) inches and (b) \(r=24\) inches.

In Exercises, find \(d y / d x\) $$ x y^{2}+4 x y=10 $$

In Exercises, find the second derivative of the function. $$ f(x)=4\left(x^{2}-1\right)^{2} $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=-x^{3}+6 x^{2}-9 x-1 $$

In Exercises, find all relative extrema of the function. $$ h(x)=-(x+4)^{3} $$

In Exercises, use the derivative to identify the open intervals on which the function is increasing or decreasing. Verify your result with the graph of the function. $$ f(x)=x^{4}-2 x^{2} $$

Let \(A\) be the area of a circle of radius \(r\) that is changing with respect to time. If \(d r / d t\) is constant, is \(d A / d t\) constant? Explain your reasoning.

In Exercises, find \(d y / d x\) $$ 4 y^{2}-x y=2 $$

In Exercises, find the second derivative of the function. $$ f(t)=\frac{3}{4 t^{2}} $$

In Exercises, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. $$ y=x^{5}+5 x^{4}-40 x^{2} $$

In Exercises, find all relative extrema of the function. $$ h(x)=2(x-3)^{3} $$

In Exercises, use the derivative to identify the open intervals on which the function is increasing or decreasing. Verify your result with the graph of the function. $$ f(x)=\frac{x^{2}}{x+1} $$

Let \(V\) be the volume of a sphere of radius \(r\) that is changing with respect to time. If \(d r / d t\) is constant, is \(d V / d t\) constant? Explain your reasoning.

In Exercises, find \(d y / d x\) $$ 2 x y^{3}-x^{2} y=2 $$

In Exercises, find the second derivative of the function. $$ g(t)=32 t^{-2} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{1}{2} x^{4}-\frac{1}{3} x^{3}-\frac{1}{2} x^{2} $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=2 x-3 $$

In Exercises, find \(d y / d x\) $$ \frac{2 y-x}{y^{2}-3}=5 $$

In Exercises, find the second derivative of the function. $$ f(x)=3\left(2-x^{2}\right)^{3} $$