Chapter 3

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$h(t)=t^{3}+3 t, \quad\quad(1,4)$$

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \sec ^{-1} x$$

Distance Let \(x\) and \(y\) be differentiable functions of \(t\) and let \(s=\sqrt{x^{2}+y^{2}}\) be the distance between the points \((x, 0)\) and \((0, y)\) in the \(x y\) -plane. a. How is \(d s / d t\) related to \(d x / d t\) if \(y\) is constant? b. How is \(d s / d t\) related to \(d x / d t\) and \(d y / d t\) if neither \(x\) nor \(y\) is constant? c. How is \(d x / d t\) related to \(d y / d t\) if \(s\) is constant?

Faster than a calculator Use the approximation \((1+x)^{k} \approx\) \(1+k x\) to estimate the following. a. \((1.0002)^{50}\) b. \(\sqrt[3]{1.009}\)

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\theta+1)-e^{\theta}$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\tan ^{3} x$$

Find \(d r / d \theta\) $$\theta^{1 / 2}+r^{1 / 2}=1$$

Find \(d y / d x\). $$f(x)=x^{3} \sin x \cos x$$

Find the derivatives of the functions. $$y=\frac{2 x+5}{3 x-2}$$

Differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function. $$y=f(x)=\frac{8}{\sqrt{x-2}}, \quad(x, y)=(6,4)$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$f(x)=\sqrt{x}, \quad\quad(4,2)$$

If \(x, y,\) and \(z\) are lengths of the edges of a rectangular box, the common length of the box's diagonals is \(s=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\). a. Assuming that \(x, y,\) and \(z\) are differentiable functions of \(t\) how is \(d s / d t\) related to \(d x / d t, d y / d t,\) and \(d z / d t ?\) b. How is \(d s / d t\) related to \(d y / d t\) and \(d z / d t\) if \(x\) is constant? c. How are \(d x / d t, d y / d t,\) and \(d z / d t\) related if \(s\) is constant?

Find the linearization of \(f(x)=\sqrt{x+1}+\sin x\) at \(x=0 .\) How is it related to the individual linearizations of \(\sqrt{x+1}\) and \(\sin x\) at \(x=0 ?\)

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=(\cos \theta) \ln (2 \theta+2)$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=5 \cos ^{-4} x$$

Find \(d r / d \theta\) $$r-2 \sqrt{\theta}=\frac{3}{2} \theta^{2 / 3}+\frac{4}{3} \theta^{3 / 4}$$

Find \(d y / d x\). $$g(x)=(2-x) \tan ^{2} x$$

Find the derivatives of the functions. $$z=\frac{4-3 x}{3 x^{2}+x}$$

Differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function. $$w=g(z)=1+\sqrt{4-z}, \quad(z, w)=(3,2)$$

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$f(x)=\sqrt{x+1}, \quad\quad(8,3)$$

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \csc ^{-1} x$$

Area The area \(A\) of a triangle with sides of lengths \(a\) and \(b\) enclosing an angle of measure \(\theta\) is $$A=\frac{1}{2} a b \sin \theta$$ a. How is \(d A / d t\) related to \(d \theta / d t\) if \(a\) and \(b\) are constant? b. How is \(d A / d t\) related to \(d \theta / d t\) and \(d a / d t\) if only \(b\) is constant? c. How is \(d A / d t\) related to \(d \theta / d t, d a / d t,\) and \(d b / d t\) if none of \(a, b,\) and \(\theta\) are constant?

Find \(d y\). $$y=x^{3}-3 \sqrt{x}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln x^{3}$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=e^{-5 x}$$

Find \(d r / d \theta\) $$\sin (r \theta)=\frac{1}{2}$$

Find \(d s / d t\). $$s=\tan t-e^{-t}$$

Find the derivatives of the functions. $$g(x)=\frac{x^{2}-4}{x+0.5}$$

Find the values of the derivatives. \(\left.\frac{d y}{d x}\right|_{x=\sqrt{3}}\) if \(y=1-\frac{1}{x}\)

Find the slope of the curve at the point indicated. $$y=5 x-3 x^{2}, \quad x=1$$

When a circular plate of metal is heated in an oven, its radius increases at the rate of \(0.01 \mathrm{cm} / \mathrm{min}\). At what rate is the plate's area increasing when the radius is \(50 \mathrm{cm} ?\)

Find \(d y\). $$y=x \sqrt{1-x^{2}}$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=e^{2 x / 3}$$

Find \(d r / d \theta\) $$\cos r+\cot \theta=e^{r \theta}$$

Find \(d s / d t\). $$s=t^{2}-\sec t+5 e^{t}$$

Find the derivatives of the functions. $$f(t)=\frac{t^{2}-1}{t^{2}+t-2}$$

Find the values of the derivatives. \(\left.\frac{d s}{d t}\right|_{t=-1} \quad\) if \(\quad s=1-3 t^{2}\) if \(\quad w=z+\sqrt{z}\)

Find the slope of the curve at the point indicated. $$y=x^{3}-2 x+7, \quad x=-2$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cos ^{-1}\left(x^{2}\right)$$

The length \(l\) of a rectangle is decreasing at the rate of \(2 \mathrm{cm} / \mathrm{sec}\) while the width \(w\) is increasing at the rate of \(2 \mathrm{cm} / \mathrm{sec} .\) When \(l=12 \mathrm{cm}\) and \(w=5 \mathrm{cm},\) find the rates of change of (a) the area, (b) the perimeter, and (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing, and which are increasing?

Find \(d y\). $$y=\frac{2 x}{1+x^{2}}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=t(\ln t)^{2}$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=e^{5-7 x}$$

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$x^{2}+y^{2}=1$$

Find \(d s / d t\). $$s=\frac{1+\csc t}{1-\csc t}$$

Find the derivatives of the functions. $$v=(1-t)\left(1+t^{2}\right)^{-1}$$

Find the values of the derivatives. \(\left.\frac{d r}{d \theta}\right|_{\theta-0}\) if \(r=\frac{2}{\sqrt{4-\theta}}\)

Find the slope of the curve at the point indicated. $$y=\frac{1}{x-1}, \quad x=3$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cos ^{-1}(1 / x)$$

Suppose that the edge lengths \(x, y,\) and \(z\) of a closed rectangular box are changing at the following rates: $$\frac{d x}{d t}=1 \mathrm{m} / \mathrm{sec}, \quad \frac{d y}{d t}=-2 \mathrm{m} / \mathrm{sec}, \quad \frac{d z}{d t}=1 \mathrm{m} / \mathrm{sec}$$ Find the rates at which the box's (a) volume, (b) surface area, and (c) diagonal length \(s=\sqrt{x^{2}+y^{2}+z^{2}}\) are changing at the instant when \(x=4, y=3,\) and \(z=2\)