Chapter 7

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{2}-1 ;(2,3) $$

Find the marginal cost for producing \(x\) units. (The cost is measured in dollars.) $$ C=205,000+9800 x $$

Find the derivative of the function. $$ g(x)=4 \sqrt[3]{x}+2 $$

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. \(f(x)=\frac{1}{x^{2}+1}\)

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=4-x^{2},(2,0) $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{2}{x-2} $$

Find the marginal cost for producing \(x\) units. (The cost is measured in dollars.) $$ C=55,000+470 x-0.25 x^{2} $$

Find the derivative of the function. $$ y=4 x^{-2}+2 x^{2} $$

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. \(f(x)=\frac{x-5}{x^{2}-9 x+20}\)

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{3}-x ;(2,6) $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{5}{x^{2}+1} $$

Find the marginal cost for producing \(x\) units. (The cost is measured in dollars.) $$ C=100(9+3 \sqrt{x}) $$

Find the derivative of the function. $$ s(t)=4 t^{-1}+1 $$

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. \(f(x)=\frac{x-1}{x^{2}+x-2}\)

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{3}+2 x ;(1,3) $$

Use the General Power Rule to find the derivative of the function. $$ y=(2 x-7)^{3} $$

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=50 x-0.5 x^{2} $$

\(f(x)=\llbracket 2 x \rrbracket+1\)

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 \sqrt{x} ;(4,4) $$

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

Use the General Power Rule to find the derivative of the function. $$ y=\left(2 x^{3}+1\right)^{2} $$

Find the derivative of the function. Use Example 7 as a model. $$ y=\frac{x^{2}-4}{x+2} $$

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=30 x-x^{2} $$

Use Example 6 as a model to find the derivative. $$ y=\frac{2}{3 x^{2}} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)-\sqrt{x+1} ;(8,3) $$

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

Use the General Power Rule to find the derivative of the function. $$ g(x)=(4-2 x)^{3} $$

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

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=-6 x^{3}+8 x^{2}+200 x $$

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. \(f(x)=\left\\{\begin{array}{ll}-2 x+3, & x<1 \\ x^{2}, & x \geq 1\end{array}\right.\)

Use the limit definition to find the derivative of the function. $$ f(x)=3 $$

Find the limit. $$ \lim _{x \rightarrow-3}(2 x+5) $$

Use the General Power Rule to find the derivative of the function. $$ h(t)=\left(1-t^{2}\right)^{4} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(t)=\left(t^{5}-1\right)\left(4 t^{2}-7 t-3\right) $$

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=50\left(20 x-x^{3 / 2}\right) $$

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. \(f(x)=\left\\{\begin{array}{ll}3+x, & x \leq 2 \\ x^{2}+1, & x>2\end{array}\right.\)

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

Find the limit. $$ \lim _{x \rightarrow 0}(3 x-2) $$

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

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

Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-2 x^{2}+72 x-145 $$

Use Example 6 as a model to find the derivative. $$ y=\frac{\sqrt{x}}{x} $$

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. \(f(x)=\left\\{\begin{array}{ll}\frac{1}{2} x+1, & x \leq 2 \\ 3-x, & x>2\end{array}\right.\)

Use the limit definition to find the derivative of the function. $$ f(x)=-5 x $$

Find the limit. $$ \lim _{x \rightarrow 1}\left(1-x^{2}\right) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(4 x-x^{2}\right)^{3} $$

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

Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-0.25 x^{2}+2000 x-1,250,000 $$

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. \(f(x)=\left\\{\begin{array}{ll}x^{2}-4, & x \leq 0 \\ 3 x+1, & x>0\end{array}\right.\)

Use the limit definition to find the derivative of the function. $$ f(x)=4 x+1 $$