Chapter 9

In Exercises 23-40, find the indicated limit. \(\lim _{x \rightarrow 2} 3\)

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

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

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{x^{3}+2 x^{2}+x-1}{x}\)

Let \(f(x)=x^{2}+6 x\). a. Find the derivative \(f^{\prime}\) of \(f\). b. Find the point on the graph of \(f\) where the tangent line to the curve is horizontal. Hint: Find the value of \(x\) for which \(f^{\prime}(x)=0\). c. Sketch the graph of \(f\) and the tangent line to the curve at the point found in part (b).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{+}} \frac{x+2}{x+1}\)

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

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

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

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=4 x^{4}-3 x^{5 / 2}+2\)

Let \(f(x)=x^{2}-2 x+1\). a, Find the derivative \(f^{\prime}\) of \(f\). b. Find the point on the graph of \(f\) where the tangent line to the curve is horizontal. c. Sketch the graph of \(f\) and the tangent line to the curve at the point found in part (b). d. What is the rate of change of \(f\) at this point?

Find the indicated limit. \(\lim _{x \rightarrow 3} x\)

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

Find the derivative of each function. \(f(x)=\frac{x+1}{2 x^{2}+2 x+3}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=5 x^{4 / 3}-\frac{2}{3} x^{3 / 2}+x^{2}-3 x+1\)

Let \(f(x)=\frac{1}{x-1}\). a. Find the derivative \(f^{\prime}\) of \(\bar{f}\) b. Find an equation of the tangent line to the curve at the point \(\left(-1,-\frac{1}{2}\right)\) c. Sketch the graph of \(f\) and the tangent line to the curve at \(\left(-1,-\frac{1}{2}\right)\).

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

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

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

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=3 x^{-1}+4 x^{-2}\)

Let \(y=f(x)=x^{2}+x\). a. Find the average rate of change of \(y\) with respect to \(x\) in the interval from \(x=2\) to \(x=3\), from \(x=2\) to \(x=2.5\), and from \(x=2\) to \(x=2.1\). b. Find the (instantaneous) rate of change of \(y\) at \(x=2\). c. Compare the results obtained in part (a) with that of part (b).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1}\)

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

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

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

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=-\frac{1}{3}\left(x^{-3}-x^{6}\right)\)

Let \(y=f(x)=x^{2}-4 x\). a. Find the average rate of change of \(y\) with respect to \(x\) in the interval from \(x=3\) to \(x=4\), from \(x=3\) to \(x=3.5\), and from \(x=3\) to \(x=3.1\). b. Find the (instantaneous) rate of change of \(y\) at \(x=3\). c. Compare the results obtained in part (a) with that of part (b).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 2^{+}} \frac{x+1}{x^{2}-2 x+3}\)

Find the indicated limit. \(\lim _{t \rightarrow 3}\left(4 t^{2}-2 t+1\right)\)

Find the derivative of each function. \(f(x)=2 x^{2}(3-4 x)^{4}\)

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

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(t)=\frac{4}{t^{4}}-\frac{3}{t^{3}}+\frac{2}{t}\)

Suppose the distance \(s\) (in feet) covered by a car moving along a straight road after \(t\) sec is given by the function \(s=f(t)=2 t^{2}+48 t\). a. Calculate the average velocity of the car over the time intervals \([20,21],[20,20.1]\), and \([20,20.01]\). b. Calculate the (instantaneous) velocity of the car when \(t=20\) c. Compare the results of part (a) with that of part (b).

Find the indicated limit. \(\lim _{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+x+2\right)\)

Find the derivative of each function. \(h(t)=t^{2}(3 t+4)^{3}\)

Find the derivative of each function. \(f(x)=\frac{x+\sqrt{3 x}}{3 x-1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{5}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x}+200\)

A ball is thrown straight up with an initial velocity of \(128 \mathrm{ft} / \mathrm{sec}\), so that its height (in feet) after \(t\) sec is given by \(s(t)=128 t-16 t^{2}\). a. What is the average velocity of the ball over the time intervals \([2,3],[2,2.5]\), and \([2,2.1] ?\) b. What is the instantaneous velocity at time \(t=2\) ? c. What is the instantaneous velocity at time \(t=5 ?\) Is the ball rising or falling at this time? d. When will the ball hit the ground?

Find the indicated limit. \(\lim _{x \rightarrow 0}\left(4 x^{5}-20 x^{2}+2 x+1\right)\)

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

In Exercises 31-34, suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=f(x) g(x)\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=2 x-5 \sqrt{x}\)

During the construction of a high-rise building, a worker accidentally dropped his portable electric screwdriver from a height of \(400 \mathrm{ft}\). After \(t\) sec, the screwdriver had fallen a distance of \(s=16 t^{2} \mathrm{ft}\). a. How long did it take the screwdriver to reach the ground? b. What was the average velocity of the screwdriver between the time it was dropped and the time it hit the ground? c. What was the velocity of the screwdriver at the time it hit the ground?

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow-2^{+}}(2 x+\sqrt{2+x})\)

Suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=\left(x^{2}+1\right) g(x)\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(t)=2 t^{2}+\sqrt{t^{3}}\)

A hot-air balloon rises vertically from the ground so that its height after \(t\) sec is \(h=\frac{1}{2} t^{2}+\frac{1}{2} t \mathrm{ft}(0 \leq t \leq 60)\). a. What is the height of the balloon at the end of \(40 \mathrm{sec}\) ? b. What is the average velocity of the balloon between \(t=0\) and \(t=40\) ? c. What is the velocity of the balloon at the end of \(40 \mathrm{sec} ?\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow-5^{+}} x(1+\sqrt{5+x})\)

Find the indicated limit. \(\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(x^{2}-4\right)\)

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