91Ó°ÊÓ

Suppose that \(N(x)\) computes the number of gallons of gas used by a vehicle traveling \(x\) miles. Suppose the vehicle gets 30 \(\mathrm{mpg.}\) a. Find a mathematical expression for \(N(x)\) b. What is \(N(100) ?\) Explain the physical meaning. c. What is \(N^{\prime}(100) ?\) Explain the physical meaning.

ITI For the function \(f(x)=x^{4}-5 x^{2}+4,\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate \(f^{\prime}(-2), f^{\prime}(-0.5), f^{\prime}(1.7),\) and \(f^{\prime}(2.718)\) .

For the function \(f(x)=x^{4}-5 x^{2}+4,\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate \(f^{\prime}(-2), f^{\prime}(-0.5), f^{\prime}(1.7),\) and \(f^{\prime}(2.718)\)

\([\mathrm{T}]\) For the function \(f(x)=\frac{x^{2}}{x^{2}+1}, \quad\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function on a graphing calculator to find \(f^{\prime}(-4), f^{\prime}(-2), f^{\prime}(2),\) and \(f^{\prime}(4)\) .

For the function \(f(x)=\frac{x^{2}}{x^{2}+1},\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function on a graphing calculator to find \(f^{\prime}(-4), f^{\prime}(-2), f^{\prime}(2),\) and \(f^{\prime}(4)\).

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=6 $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=6$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=2-3 x $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=2-3 x$$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\frac{2 x}{7}+1$$