Chapter 4

Evaluate \(\int \frac{1}{|x| \sqrt{x^{2}-1}} d x\) by rewriting the integrand as \(\frac{1}{x^{2} \sqrt{1-1 / x^{2}}}\) and then making the substitution \(u=1 / x .\) Use your answer to derive an identity involving \(\sin ^{-1}(1 / x)\) and \(\sec ^{-1} x\)

Involve the just-in-time inventory discussed in the chapter introduction. A further refinement we can make to the EOQ model of exercises \(62-63\) is to allow discounts for ordering large quantities. To make the calculations easier, take specific values of \(D=4000, C_{o}=\$ 50,000\) and \(C_{c}=\$ 3800 .\) If \(1-99\) items are ordered, the price is \(\$ 2800\) per item. If \(100-179\) items are ordered, the price is \(\$ 2200\) per item. If 180 or more items are ordered, the price is \(\$ 1800\) per item. The total cost is now \(C_{o} \frac{D}{Q}+C_{c} \frac{Q}{2}+P D,\) where \(P\) is the price per item. Find the order size \(Q\) that minimizes the total cost.

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int x^{2} \sqrt{x^{3}+2} d x$$

The location \((\bar{x}, \bar{y})\) of the center of gravity (balance point) of a flat plate bounded by \(y=f(x)>0, a \leq x \leq b\) and the \(x\) -axis is given by \(\bar{x}=\frac{\int_{a}^{b} x f(x) d x}{\int_{a}^{b} f(x) d x}\) and \(\bar{y}=\frac{\int_{a}^{b}[f(x)]^{2} d x}{2 \int_{a}^{b} f(x) d x} .\) For the semicircle \(y=f(x)=\sqrt{4-x^{2}},\) use symmetry to argue that \(\bar{x}=0\) and \(\bar{y}=\frac{1}{2 \pi} \int_{0}^{2}\left(4-x^{2}\right) d x .\) Compute \(\bar{y}\)

The impulse-momentum equation states the relationship between a force \(F(t)\) applied to an object of mass \(m\) and the resulting change in velocity \(\Delta v\) of the object. The equation is \(m \Delta v=\int_{a}^{b} F(t) d t,\) where \(\Delta v=v(b)-v(a) .\) Suppose that the force of a baseball bat on a ball is approximately \(F(t)=9-10^{8}(t-0.0003)^{2}\) thousand pounds, for \(t\) between 0 and 0.0006 second. What is the maximum force on the ball? Using \(m=0.01\) for the mass of a baseball, estimate the change in velocity \(\Delta v\) (in \(\mathrm{f} t / \mathrm{s}\) ).

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int\left(x \sin 2 x+x^{2} \cos 2 x\right) d x$$

Suppose that the population density of a group of animals can be described by \(f(x)=x e^{-x^{2}}\) thousand animals per mile for \(0 \leq x \leq 2,\) where \(x\) is the distance from a pond. Graph \(y=f(x)\) and brictly describe where these animals are likely to be found. Find the total population \(\int_{0}^{2} f(x) d x\)

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int \frac{2 x e^{3 x}-3 x^{2} e^{3 x}}{e^{6 x}} d x$$

The voltage in an AC (alternating current) circuit is given by \(V(t)=V_{p} \sin (2 \pi f t),\) where \(f\) is the frequency. A voltmeter does not indicate the amplitude \(V_{p} .\) Instead, the voltmeter reads the root-mean-square (rms), the square root of the average value of the square of the voltage over one cycle. That is, \(\mathrm{rms}=\sqrt{f \int_{0}^{1 / f} V^{2}(t) d t} .\) Use the trigonometric identity \(\sin ^{2} x=\frac{1}{2}-\frac{1}{2} \cos 2 x\) to show that \(\mathrm{rms}=V_{p} / \sqrt{2}\)

Let \(f(x)=\left\\{\begin{array}{cl}x & \text { if } x<2 \\ x+1 & \text { if } x \geq 2\end{array} \text { and define } F(x)=\int_{0}^{x} f(t) d t\right.\) Show that \(F(x)\) is continuous but that it is not true that \(F^{\prime}(x)=f(x)\) for all \(x .\) Explain why this does not contradict the Fundamental Theorem of Calculus.

Use a graph to explain why \(\int_{-1}^{1} x^{3} d x=0 .\) Use your knowledge of \(e^{-x}\) to determine whether \(\int_{-1}^{1} x^{3} e^{-x} d x\) is positive or negative.

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int \frac{x \cos x^{2}}{\sqrt{\sin x^{2}}} d x$$

Graph \(y=f(t)\) and find the root-mean-square of $$f(t)=\left\\{\begin{array}{ll} -1 & \text { if }-2 \leq t<-1 \\ t & \text { if }-1 \leq t \leq 1 \\ 1 & \text { if } 1

Find the derivative of \(f(x)=\frac{1}{k} \int_{x}^{x+k} g(t) d t,\) where \(g\) is a continuous function.

Use the Integral Mean Value Theorem to prove the following fact for a continuous function. For any positive integer \(n\), there exists a set of evaluation points for which the Riemann sum approximation of \(\int_{a}^{b} f(x) d x\) is exact.

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int\left(\sqrt{x^{2}+1} \cos x+\frac{x}{\sqrt{x^{2}+1}} \sin x\right) d x$$

Find first and second derivatives of \(g(x)=\int_{0}^{x}\left(\int_{0}^{u} f(t) d t\right) d u,\) where \(f\) is a continuous function. Identify the graphical feature of \(y=g(x)\) that corresponds to a zero of \(f(x)\)

Show that \(\int \frac{-1}{\sqrt{1-x^{2}}} d x=\cos ^{-1} x+c\) and \(\int \frac{-1}{\sqrt{1-x^{2}}} d x=-\sin ^{-1} x+c\) Explain why this does not imply that \(\cos ^{-1} x=-\sin ^{-1} x .\) Find an equation relating \(\cos ^{-1} x\) and \(\sin ^{-1} x\)

Let \(f\) be a continuous function on the interval \([0,1],\) and \(\mathrm{de}-\) fine \(g_{n}(x)=f\left(x^{n}\right)\) for \(n=1,2\) and so on. For a given \(x\) with \(0 \leq x \leq 1,\) find \(\lim _{n \rightarrow \infty} g_{n}(x) .\) Then, find \(\lim _{n \rightarrow \infty} \int_{0}^{1} g_{n}(x) d x\)

Derive the formulas \(\int \sec ^{2} x d x=\tan x+c\) and \(\int \sec x \tan x \, d x=\sec x+c\)

Identify the integrals to which the Fundamental Theorem of Calculus applies; the other integrals are called improper integrals. (a) \(\int_{0}^{4} \frac{1}{x-4} d x\) (b) \(\int_{0}^{1} \sqrt{x} d x\) (c) \(\int_{0}^{1} \ln x d x\)

Derive the formulas \(\int e^{x} d x=e^{x}+c\) and \(\int e^{-x} d x=-e^{-x}+c\)

Identify the integrals to which the Fundamental Theorem of Calculus applies; the other integrals are called improper integrals. (a) \(\int_{0}^{1} \frac{1}{\sqrt{x+2}} d x \quad\) (b) \(\int_{0}^{2} \frac{1}{(x-3)^{2}} d x \quad\) (c) \(\int_{0}^{2} \sec x d x\)

Identify each sum as a Riemann sum and evaluate the limit. (a) \(\lim _{n \rightarrow \infty} \frac{1}{n}\left[\sin \frac{\pi}{n}+\sin \frac{2 \pi}{n}+\cdots+\sin \pi\right]\) (b) \(\lim _{n \rightarrow \infty} \frac{2}{n}\left[\frac{1}{1+2 / n}+\frac{1}{1+4 / n}+\cdots+\frac{1}{3}\right]\)

Identify each sum as a Riemann sum and evaluate the limit. (a) \(\lim _{n \rightarrow \infty} \frac{1}{n}\left[e^{4 / n}+e^{8 / n}+\cdots+e^{4}\right]\) (b) \(\lim _{n \rightarrow \infty} \frac{4}{n}\left[\frac{2}{\sqrt{n}}+\frac{2 \sqrt{2}}{\sqrt{n}}+\cdots+2\right]\)

Derive Leibniz' Rule: $$\frac{d}{d x} \int_{a(x)}^{b(x)} f(t) d t=f(b(x)) b^{\prime}(x)-f(a(x)) a^{\prime}(x)$$