Chapter 7

Find the limit. $$ \lim _{x \rightarrow 2}\left(-x^{2}+x-2\right) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(x^{2}-9\right)^{2 / 3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\sqrt[3]{x}(\sqrt{x}+3) $$

Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-0.00025 x^{2}+12.2 x-25,000 $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{|x+1|}{x+1}\)

Use the limit definition to find the derivative of the function. $$ g(s)=\frac{1}{3} s+2 $$

Find the limit. $$ \lim _{x \rightarrow 3} \sqrt{x+6} $$

Use the General Power Rule to find the derivative of the function. $$ f(t)=(9 t+2)^{2 / 3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\sqrt[3]{x}(x+1) $$

Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-0.5 x^{3}+30 x^{2}-164.25 x-1000 $$

Find the value of the derivative of the function at the given point. $$ f(t)=4-\frac{4}{3 t} \quad\left(\frac{1}{2}, \frac{4}{3}\right) $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{|4-x|}{4-x}\)

Use the limit definition to find the derivative of the function. $$ h(t)=6-\frac{1}{2} t $$

Find the limit. $$ \lim _{x \rightarrow 4} \sqrt[3]{x+4} $$

Use the General Power Rule to find the derivative of the function. $$ f(t)=\sqrt{t+1} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{3 x-2}{2 x-3} $$

The cost \(C\) (in dollars) of producing \(x\) units of a product is given by \(C=3.6 \sqrt{x}+500\) (a) Find the additional cost when the production increases from 9 to 10 units. (b) Find the marginal cost when \(x=9\). (c) Compare the results of parts (a) and (b).

Find the value of the derivative of the function at the given point. $$ f(x)=-\frac{1}{2} x\left(1+x^{2}\right) \quad(1,-1) $$

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\llbracket x-1 \rrbracket\)

Use the limit definition to find the derivative of the function. $$ f(x)=x^{2}-4 $$

Find the limit. $$ \lim _{x \rightarrow-3} \frac{2}{x+2} $$

Use the General Power Rule to find the derivative of the function. $$ g(x)=\sqrt{5-3 x} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{x^{3}+3 x+2}{x^{2}-1} $$

The revenue \(R\) (in dollars) from renting \(x\) apartments can be modeled by \(R=2 x\left(900+32 x-x^{2}\right)\) (a) Find the additional revenue when the number of rentals is increased from 14 to 15 . (b) Find the marginal revenue when \(x=14\). (c) Compare the results of parts (a) and (b).

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=x-\llbracket x \rrbracket\)

Use the limit definition to find the derivative of the function. $$ f(x)=1-x^{2} $$

Find the limit. $$ \lim _{x \rightarrow-2} \frac{3 x+1}{2-x} $$

Use the General Power Rule to find the derivative of the function. $$ s(t)=\sqrt{2 t^{2}+5 t+2} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{3-2 x-x^{2}}{x^{2}-1} $$

The profit \(P\) (in dollars) from selling \(x\) units of calculus textbooks is given by \(P=-0.05 x^{2}+20 x-1000\) (a) Find the additional profit when the sales increase from 150 to 151 units. (b) Find the marginal profit when \(x=150\). (c) Compare the results of parts (a) and (b).

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(h(x)=f(g(x)), \quad f(x)=\frac{1}{\sqrt{x}}, \quad g(x)=x-1, x>1\)

Use the limit definition to find the derivative of the function. $$ h(t)=\sqrt{t-1} $$

Find the limit. $$ \lim _{x \rightarrow-2} \frac{x^{2}-1}{2 x} $$

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{3 x^{3}+4 x} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\left(x^{5}-3 x\right)\left(\frac{1}{x^{2}}\right) $$

The population \(P\) (in thousands) of Japan can be modeled by \(P=-14.71 t^{2}+785.5 t+117,216\) where \(t\) is time in years, with \(t=0\) corresponding to 1980 . (a) Evaluate \(P\) for \(t=0,10,15,20\), and 25 . Explain these values. (b) Determine the population growth rate, \(d P / d t\). (c) Evaluate \(d P / d t\) for the same values as in part (a). Explain your results.

Use the limit definition to find the derivative of the function. $$ f(x)=\sqrt{x+2} $$

Find the limit. $$ \lim _{x \rightarrow-1} \frac{4 x-5}{3-x} $$

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{9 x^{2}+4} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x\left(1-\frac{2}{x+1}\right) $$

The temperature \(T\) (in degrees Fahrenheit) of a person during an illness can be modeled by the equation \(T=-0.0375 t^{2}+0.3 t+100.4\), where \(t\) is time in hours since the person started to show signs of a fever. (a) Use a graphing utility to graph the function. Be sure to choose an appropriate window. \(3 !\) (b) Do the slopes of the tangent lines appear to be positive or negative? What does this tell you? (c) Evaluate the function for \(t=0,4,8\), and \(12 .\) (d) Find \(d T / d t\) and explain its meaning in this situation. (e) Evaluate \(d T / d t\) for \(t=0,4,8\), and \(12 .\)

find \(f^{\prime}(x)\). $$ f(x)=x^{2}-\frac{4}{x}-3 x^{-2} $$

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=x^{2}-4 x-5\)

Use the limit definition to find the derivative of the function. $$ f(t)=t^{3}-12 t $$

Find the limit. $$ \lim _{x \rightarrow 7} \frac{5 x}{x+2} $$

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(t)=\frac{t+2}{t^{2}+5 t+6} $$

The profit \(P\) (in dollars) from selling \(x\) units of a product is given by \(P=36,000+2048 \sqrt{x}-\frac{1}{8 x^{2}}, \quad 150 \leq x \leq 275\) Find the marginal profit for each of the following sales. (a) \(x=150\) (b) \(x=175\) (c) \(x=200\) (d) \(x=225\) (e) \(x=250\) (f) \(x=275\)

find \(f^{\prime}(x)\). $$ f(x)=x^{2}-3 x-3 x^{-2}+5 x^{-3} $$

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{5}{x^{2}+1} \quad[-2,2]\)