Chapter 9

Find the derivative of the function. \(g(t)=\frac{(2 t-1)^{2}}{(3 t+2)^{4}}\)

In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\frac{1+2 x^{1 / 2}}{1+x^{3 / 2}} ;\left(4, \frac{5}{9}\right)\)

Find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=-\frac{5}{3} x^{2}+2 x+2 ;\left(-1,-\frac{5}{3}\right)\)

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes time and \(f(x)\) denotes the prime interest rate at time \(x\).

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=|x-1| $$

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} 2 f(x)\)

Find the derivative of the function. \(f(x)=\frac{\sqrt{2 x+1}}{x^{2}-1}\)

In Exercises 43-48, find the first and second derivatives of the given function. \(f(x)=4 x^{2}-2 x+1\)

Find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=x^{4}-3 x^{3}+2 x^{2}-x+1 ;(1,0)\)

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes time and \(f(x)\) denotes a country's industrial production.

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} x+5 & \text { if } x<0 \\ 2 & \text { if } x=0 \\ -x^{2}+5 & \text { if } x>0 \end{array}\right. $$

Find the derivative of the function. \(f(t)=\frac{4 t^{2}}{\sqrt{2 t^{2}+2 t-1}}\)

Find the first and second derivatives of the given function. \(f(x)=-0.2 x^{2}+0.3 x+4\)

Find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\sqrt{x}+\frac{1}{\sqrt{x}} ;\left(4, \frac{5}{2}\right)\)

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes the level of production of a certain commodity, and \(f(x)\) denotes the total cost incurred in producing \(x\) units of the commodity.

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} \frac{x^{2}-1}{x+1} & \text { if } x \neq-1 \\ 1 & \text { if } x=-1 \end{array}\right. $$

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a}[f(x) g(x)]\)

Find the derivative of the function. \(g(t)=\frac{\sqrt{t+1}}{\sqrt{t^{2}+1}}\)

Find the first and second derivatives of the given function. \(f(x)=2 x^{3}-3 x^{2}+1\)

Let \(f(x)=x^{3}\). a. Find the point on the graph of \(f\) where the tangent line is horizontal. b. Sketch the graph of \(f\) and draw the horizontal tangent line.

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes altitude and \(f(x)\) denotes atmospheric pressure.

In Exercises 45-56, find the values of \(x\) for which each function is continuous. \(f(x)=2 x^{2}+x-1\)

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} \sqrt{g(x)}\)

Find the derivative of the function.\(f(x)=\frac{\sqrt{x^{2}+1}}{\sqrt{x^{2}-1}}\)

Find the first and second derivatives of the given function. \(g(x)=-3 x^{3}+24 x^{2}+6 x-64\)

Let \(f(x)=x^{3}-4 x^{2}\). Find the point(s) on the graph of \(f\) where the tangent line is horizontal.

Find the values of \(x\) for which each function is continuous. \(f(x)=x^{3}-2 x^{2}+x-1\)

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} \sqrt[3]{5 f(x)+3 g(x)}\)

Find the derivative of the function. \(f(x)=(3 x+1)^{4}\left(x^{2}-x+1\right)^{3}\)

Find the first and second derivatives of the given function. \(h(t)=t^{4}-2 t^{3}+6 t^{2}-3 t+10\)

Let \(f(x)=x^{3}+1\). a. Find the point(s) on the graph of \(f\) where the slope of the tangent line is equal to 12 . b. Find the equation(s) of the tangent line(s) of part (a). c. Sketch the graph of \(f\) showing the tangent line(s).

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{2}{x^{2}+1}\)

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)}\)

Find the first and second derivatives of the given function. \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-1\)

Let \(f(x)=\frac{2}{3} x^{3}+x^{2}-12 x+6\). Find the values of \(x\) for which: a. \(f^{\prime}(x)=-12\) b. \(f^{\prime}(x)=0\) c. \(f^{\prime}(x)=12\)

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{x}{2 x^{2}+1}\)

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} \frac{g(x)-f(x)}{f(x)+\sqrt{g(x)}}\)

In Exercises 49-54, find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=u^{4 / 3}\) and \(u=3 x^{2}-1\)

In Exercises 49-52, find the third derivative of the given function. \(f(x)=3 x^{4}-4 x^{3}\)

Let \(f(x)=\frac{1}{4} x^{4}-\frac{1}{3} x^{3}-x^{2} .\) Find the point(s) on the graph of \(f\) where the slope of the tangent line is equal to: a. \(-2 x\) b. 0 c. \(10 x\)

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{2}{2 x-1}\)

In Exercises 49-62, find the indicated limit, if it exists. \(\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}\)

Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=\sqrt{u}\) and \(u=7 x-2 x^{2}\)

Find the third derivative of the given function. \(f(x)=3 x^{5}-6 x^{4}+2 x^{2}-8 x+12\)

A straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve. Find an equation of the tangent line and the normal to the curve \(y=x^{3}-3 x+1\) at the point \((2,3)\).

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{x+1}{x-1}\)

Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=u^{-2 / 3}\) and \(u=2 x^{3}-x+1\)

Find the third derivative of the given function. \(f(x)=\frac{1}{x}\)

Gnown of A CaNcenous Tumon The volume of a spherical cancerous tumor is given by the function $$ V(r)=\frac{4}{3} \pi r^{3} $$ where \(r\) is the radius of the tumor in centimeters. Find the rate of change in the volume of the tumor when a. \(r=\frac{2}{3} \mathrm{~cm}\) b. \(r=\frac{5}{4} \mathrm{~cm}\)

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{2 x+1}{x^{2}+x-2}\)