Chapter 3

Confirm that the formula is the local linear approximation at \(x_{0}=0,\) and use a graphing utility to estimate an interval of \(x\) -values on which the error is at most ±0.1. $$\frac{1}{(1+2 x)^{5}} \approx 1-10 x$$

Find \(d y / d x\). $$y=\sin \left(e^{x}\right)$$

Find the slope of the tangent line to the curve at the given points in two ways: first by solving for \(y\) in terms of \(x\) and differentiating and then by implicit differentiation. $$y^{2}-x+1=0 ;(10,3),(10,-3)$$

Find \(d y / d x\). $$y=\ln (\ln (\ln x))$$

Find the limits. $$\lim _{x \rightarrow+\infty} x e^{-x}$$

Determine whether the statement is true or false. Explain your answer. If an equation in \(x\) and \(y\) defines a function \(y=f(x)\) implicitly, then the graph of the equation and the graph of \(f\) are identical.

Find \(d y / d x\). $$y=e^{x \tan x}$$

Find \(d y / d x\). $$y=\ln (\tan x)$$

Find the limits. $$\lim _{x \rightarrow \pi^{-}}(x-\pi) \tan \frac{1}{2} x$$

(a) Use the local linear approximation of \(\tan x\) at \(x_{0}=0\) to approximate tan \(2^{\circ},\) and compare the approximation to the result produced directly by your calculating device. (b) How would you choose \(x_{0}\) to approximate \(\tan 61^{\circ} ?\) (c) Approximate \(\tan 61^{\circ} ;\) compare the approximation to the result produced directly by your calculating device.

Determine whether the statement is true or false. Explain your answer. The function $$f(x)=\left\\{\begin{array}{rr}\sqrt{1-x^{2}}, & 0

Find \(d y / d x\). $$y=\frac{e^{x}}{\ln x}$$

Find \(d y / d x\). $$y=\ln (\cos x)$$

Find the limits. $$\lim _{x \rightarrow+\infty} x \sin \frac{\pi}{x}$$

Use an appropriate local linear approximation to estimate the value of the given quantity. $$(3.02)^{4}$$

Determine whether the statement is true or false. Explain your answer. The function \(|x|\) is not defined implicitly by the equation \((x+y)(x-y)=0\).

Find \(d y / d x\). $$y=e^{\left(x-e^{3 x}\right)}$$

Find \(d y / d x\). $$y=\cos (\ln x)$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} \tan x \ln x$$

Use an appropriate local linear approximation to estimate the value of the given quantity. $$(1.97)^{3}$$

Determine whether the statement is true or false. Explain your answer. If \(y\) is defined implicitly as a function of \(x\) by the equation \(x^{2}+y^{2}=1,\) then \(d y / d x=-x / y\).

Find \(d y / d x\). $$y=\exp (\sqrt{1+5 x^{3}})$$

Find \(d y / d x\). $$y=\sin ^{2}(\ln x)$$

Find the limits. $$\lim _{x \rightarrow \pi / 2^{-}} \sec 3 x \cos 5 x$$

A conical water tank with vertex down has a radius of \(10 \mathrm{ft}\) at the top and is \(24 \mathrm{ft}\) high. If water flows into the tank at a rate of \(20 \mathrm{ft}^{3} / \mathrm{min}\), how fast is the depth of the water increasing when the water is 16 ft deep?

Use implicit differentiation to find the slope of the tangent line to the curve at the specified point, and check that your answer is consistent with the accompanying graph. \(x^{4}+y^{4}=16 ;(1, \sqrt[4]{15}) \quad[\text { Lamé's special quartic }]\)

Find \(d y / d x\). $$y=\ln \left(1-x e^{-x}\right)$$

Find \(d y / d x\). $$y=\log \left(\sin ^{2} x\right)$$

Find the limits. $$\lim _{x \rightarrow \pi}(x-\pi) \cot x$$

Grain pouring from a chute at the rate of \(8 \mathrm{ft}^{3} / \mathrm{min}\) forms a conical pile whose height is always twice its radius. How fast is the height of the pile increasing at the instant when the pile is \(6 \mathrm{ft}\) high?

Use an appropriate local linear approximation to estimate the value of the given quantity. $$\sqrt{24}$$

Find \(d y / d x\). $$y=\ln \left(\cos e^{x}\right)$$

Find \(d y / d x\). $$y=\log \left(1-\sin ^{2} x\right)$$

Find the limits. $$\lim _{x \rightarrow+\infty}(1-3 / x)^{x}$$

Sand pouring from a chute forms a conical pile whose height is always equal to the diameter. If the height increases at a constant rate of \(5 \mathrm{ft} / \mathrm{min}\), at what rate is sand pouring from the chute when the pile is 10 ft high?

Use implicit differentiation to find the slope of the tangent line to the curve at the specified point, and check that your answer is consistent with the accompanying graph. \(2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right) ;(3,1) \quad[\text { lemniscate }]\)

Use the method of Example 3 to help perform the indicated differentiation. $$\frac{d}{d x}\left[\ln \left((x-1)^{3}\left(x^{2}+1\right)^{4}\right)\right]$$

Find the limits. $$\lim _{x \rightarrow 0}(1+2 x)^{-3 / x}$$

Wheat is poured through a chute at the rate of \(10 \mathrm{ft}^{3} / \mathrm{min}\) and falls in a conical pile whose bottom radius is always half the altitude. How fast will the circumference of the base be increasing when the pile is 8 ft high?

Use an appropriate local linear approximation to estimate the value of the given quantity. $$\sqrt{36.03}$$

Use implicit differentiation to find the slope of the tangent line to the curve at the specified point, and check that your answer is consistent with the accompanying graph. \(x^{2 / 3}+y^{2 / 3}=4 ;(-1,3 \sqrt{3}) \quad\) [four-cusped hypocycloid]

Find the limits. $$\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / x}$$

An aircraft is climbing at a \(30^{\circ}\) angle to the horizontal. How fast is the aircraft gaining altitude if its speed is \(500 \mathrm{mi} / \mathrm{h} ?\)

Use an appropriate local linear approximation to estimate the value of the given quantity. $$\sin 0.1$$

Use implicit differentiation to find the specified derivative. $$a^{4}-t^{4}=6 a^{2} t ; d a / d t$$

Find the limits. $$\lim _{x \rightarrow+\infty}(1+a / x)^{b x}$$

Use implicit differentiation to find the specified derivative. $$\sqrt{u}+\sqrt{v}=5 ; \quad d u / d v$$

Find \(f^{\prime}(x)\) by Formula (7) and then by logarithmic differentiation. $$f(x)=\pi^{x \tan x}$$

Find the limits. $$\lim _{x \rightarrow 1}(2-x)^{\tan [(\pi / 2) x]}$$

Use implicit differentiation to find the specified derivative. $$a^{2} \omega^{2}+b^{2} \lambda^{2}=1(a, b \text { constants }) ; \quad d \omega / d \lambda$$