Chapter 2: Problem 26
Volume of Sphere What is the rate of change of the volume of a sphere with respect to the radius when the radius is \(r=2\) in.?
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In Exercises \(15-18\) , determine whether the curve has a tangent at the indicated point, If it does, give its slope, If not, explain why not. $$f(x)=\left\\{\begin{array}{ll}{1 / x,} & {x \leq 2} \\ {\frac{4-x}{4},} & {x>2}\end{array}\right.\( at \)x=2$$
In Exercises 49 and 50 , determine the limit. Assume that $$\lim _ { x \rightarrow 4 } f ( x ) = 0$$ and $$\lim _ { x \rightarrow 4 } g ( x ) = 3$$ (a) $$\lim _ { x \rightarrow 4 } ( g ( x ) + 3 )$$ (b) $$\lim _ { x \rightarrow 4 } x f ( x )$$ (c) $$\lim _ { x \rightarrow 4 } g ^ { 2 } ( x ) \quad \quad$$ (d) $$\lim _ { x \rightarrow 4 } \frac { g ( x ) } { f ( x ) - 1 }$$
In Exercises \(71 - 74 ,\) complete the following tables and state what you believe \(\lim _ { x \rightarrow 0 } f ( x )\) to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = x \sin \frac { 1 } { x }$$
In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=2+\cos t\) (a) \([0, \pi]\) \((\mathbf{b})[-\pi, \pi]\)
In Exercises \(19-22,\) (a) find the slope of the curve at \(x=a\) . (b) Writing to Learn Describe what happens to the tangent at \(x=a\) as \(a\) changes. $$y=9-x^{2}$$
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