Chapter 4

Use the following definitions. The upper sum of \(f\) on \(P\) is given by \(U(P, f)=\sum_{i=1}^{n} f\left(c_{i}\right) \Delta x,\) where \(f\left(c_{i}\right)\) is the maximum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right] .\) Similarly, the lower sum of \(f\) on \(P\) is given by \(L(P, f)=\sum_{i=1}^{m} f\left(d_{i}\right) \Delta x,\) where \(f\left(d_{i}\right)\) is the minimum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right]\) Find (a) the general upper sum and (b) the general lower sum for \(f(x)=x^{2}\) on [0,2] and show that both sums approach the same number as \(n \rightarrow \infty\)

Compute the average value of the function on the given interval. $$f(x)=x^{2}-1,[1,3]$$

Use a CAS to find an antiderivative, then verify the answer by computing a derivative. (a) \(\int x^{2} e^{-x^{3}} d x\) (b) \(\int \frac{1}{x^{2}-x} d x\) (c) \(\int \csc x \, d x\)

Evaluate the definite integral. $$\int_{1}^{\varepsilon} \frac{\ln x}{x} d x$$

Starting with \(\quad e^{x}=\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}, \quad\) show \(\quad\) that \(\ln x=\lim _{n \rightarrow \infty}\left[n\left(x^{1 / n}-1\right)\right] .\) Assume that if \(\lim _{n \rightarrow \infty} x_{n}=x,\) then \(\lim _{n \rightarrow \infty}\left[n\left(x_{n}^{1 / n}-1\right)\right]=\lim _{n \rightarrow \infty}\left[n\left(x^{1 / n}-1\right)\right].\)

Katie drives a car at speed \(f(t)=55+10 \cos t\) mph, and Michael drives a car at speed \(g(t)=50+2 t\) mph at time \(t\) minutes. Suppose that Katie and Michael are at the same location at time \(t=0 .\) Compute \(\int_{0}^{x}[f(t)-g(t)] d t,\) and interpret the integral in terms of a race between Katie and Michael.

Compute the average value of the function on the given interval. $$f(x)=2 x-2 x^{2},[0,1]$$

Use a CAS to find an antiderivative, then verify the answer by computing a derivative. (a) \(\int \frac{x}{x^{4}+1} d x\) (b) \(\int 3 x \sin 2 x \, d x\) (c) \(\int \ln x \, d x\)

Evaluate the definite integral. $$\int_{1}^{4} \frac{x-1}{\sqrt{x}} d x$$

The sigmoid function \(f(x)=\frac{1}{1+e^{-x}}\) is used to model situations with a threshold. For example, in the brain, each neuron receives inputs from numerous other neurons and fires only after its total input crosses some threshold. Graph \(y=f(x)\) and find \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow \infty} f(x) .\) Define the function \(g(x)\) to be the value of \(f(x)\) rounded off to the nearest integer. What value of \(x\) is the threshold for this function to switch from "off" (0) to "on" (1)? How could you modify the function to move the threshold to \(x=4\) instead?

Find the given area. The area above the \(x\) -axis and below \(y=4-x^{2}\)

Prove that if \(f\) is continuous on the interval \([a, b],\) then there exists a number \(c\) in \((a, b)\) such that \(f(c)\) equals the average value of \(f\) on the interval \([a, b]\).

Find the function \(f(x)\) satisfying the given conditions. $$f^{\prime}(x)=3 e^{x}+x, f(0)=4$$

Evaluate the definite integral. $$\int_{0}^{1} \frac{x}{\sqrt{x^{2}+1}} d x$$

Suppose you have a 1 -in- 10 chance of winning a prize with some purchase (like a lottery). If you make 10 purchases (i.e., you get 10 tries), the probability of winning at least one prize is \(1-(9 / 10)^{10} .\) If the prize had probability 1 -in-20 and you tried 20 times, would the probability of winning at least once be higher or lower? Compare \(1-(9 / 10)^{10}\) and \(1-(19 / 20)^{20}\) to find out. To see what happens for larger and larger odds, compute \(\lim _{n \rightarrow \infty}\left[1-((n-1) / n)^{n}\right].\)

Find the given area. The area below the \(x\) -axis and above \(y=x^{2}-4 x\)

Find the function \(f(x)\) satisfying the given conditions. $$f^{\prime}(x)=4 \cos x, f(0)=3$$

Use induction to derive the geometric series formula \(a+a r+a r^{2}+\cdots+a r^{n}=\frac{a-a r^{n+1}}{1-r}\) for constants \(a\) and \(r \neq 1.\)

Evaluate the integral exactly, if possible. Otherwise, estimate it numerically. (a) \(\int_{0}^{\pi} \sin x^{2} d x\) (b) \(\int_{0}^{\pi} x \sin x^{2} d x\)

In the text, we deferred the proof of \(\lim _{h \rightarrow \infty} \frac{e^{h}-1}{h}=1\) to the exercises. In this exercise, we guide you through one possible proof. (Another proof is given in exercise 42.) Starting with \(h>0,\) write \(h=\ln e^{h}=\int_{1}^{e^{h}} \frac{1}{x} d x .\) Use the Integral Mean Value Theorem to write \(\int_{1}^{e^{h}} \frac{1}{x} d x=\frac{e^{h}-1}{\bar{x}}\) for some number \(\bar{x}\) between 1 and \(e^{h} .\) This gives you \(\frac{e^{h}-1}{h}=\bar{x} .\) Now, take the limit as \(h \rightarrow 0^{+} .\) For \(h<0,\) repeat this argument, with \(h\) replaced with \(-h.\)

Find the given area. The area of the region bounded by \(y=x^{2}, x=2\) and the \(x\) -axis

Find the function \(f(x)\) satisfying the given conditions. $$f^{\prime \prime}(x)=12, f^{\prime}(0)=2, f(0)=3$$

Evaluate the integral exactly, if possible. Otherwise, estimate it numerically. (a) \(\int_{-1}^{1} x e^{-x^{2}} d x\) (b) \(\int_{-1}^{1} e^{-x^{2}} d x\)

In this exercise, we guide you through a different proof of \(\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=1 .\) Start with \(f(x)=\ln x\) and the fact that \(f^{\prime}(1)=1 .\) Using the alternative definition of derivative, we write this as \(f^{\prime}(1)=\lim _{x \rightarrow 1} \frac{\ln x-\ln 1}{x-1}=1 .\) Explain why this implies that \(\lim _{x \rightarrow 1} \frac{x-1}{\ln x}=1 .\) Finally, substitute \(x=e^{h}.\)

Find the given area. The area of the region bounded by \(y=x^{3}, x=3\) and the \(x\) -axis

Find the function \(f(x)\) satisfying the given conditions. $$f^{\prime \prime}(x)=2 x, f^{\prime}(0)=-3, f(0)=2$$

Evaluate the integral exactly, if possible. Otherwise, estimate it numerically. (a) \(\int_{0}^{2} \frac{4 x^{2}}{\left(x^{2}+1\right)^{2}} d x\) (b) \(\int_{0}^{2} \frac{4 x^{3}}{\left(x^{2}+1\right)^{2}} d x\)

In this exercise, we show that if \(e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n},\) then \(\ln e=1 .\) Define \(x_{n}=\left(1+\frac{1}{n}\right)^{n} .\) By the continuity of \(\ln x\) we have \(\ln e=\ln \left[\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}\right]=\lim _{n \rightarrow \infty}\left[\ln \left(1+\frac{1}{n}\right)^{n}\right].\) Use I'Hôpital's Rule on \(\lim _{n \rightarrow \infty}\left[\frac{\ln \left(1+\frac{1}{n}\right)}{1 / n}\right]\) to evaluate this limit.

Suppose that \(R_{L}\) and \(R_{R}\) are the Riemann sum approximations of \(\int_{a}^{b} f(x) d x\) using left- and right-endpoint evaluation rules, respectively, for some \(n > 0 .\) Show that the trapezoidal approximation \(T_{n}\) is equal to \(\left(R_{L}+R_{R}\right) / 2\)

Find the given area. The area between \(y=\sin x\) and the \(x\) -axis for \(0 \leq x \leq \pi\)

Find all functions satisfying the given conditions. $$f^{\prime \prime}(x)=3 \sin x+4 x^{2}$$

Evaluate the integral exactly, if possible. Otherwise, estimate it numerically. (a) \(\int_{0}^{\pi / 4} \sec x d x\) (b) \(\int_{0}^{\pi / 4} \sec ^{2} x d x\)

Apply Newton's method to the function \(f(x)=\ln x-1\) to find an iterative scheme for approximating \(e .\) Discover how many steps are needed to start at \(x_{0}=3\) and obtain five digits of accuracy.

Find the given area. The area between \(y=\sin x \quad\) and the \(\quad x\) -axis for \(-\pi / 2 \leq x \leq \pi / 4\)

Find all functions satisfying the given conditions. $$f^{\prime \prime}(x)=\sqrt{x}-2 \cos x$$

Make the indicated substitution for an unspecified function \(f(x)\). $$u=x^{2} \text { for } \int_{0}^{2} x f\left(x^{2}\right) d x$$

A telegraph cable is made of an outer winding around an inner core. If \(x\) is defined as the core radius divided by the outer radius, the transmission speed is proportional to \(s(x)=x^{2} \ln (1 / x) .\) Find an \(x\) that maximizes the transmission speed.

Show that both \(\int_{0}^{1} \sqrt{1-x^{2}} d x\) and \(\int_{0}^{1} \frac{1}{1+x^{2}} d x\) equal \(\frac{\pi}{4}\) Use Simpson's Rule on each integral with \(n=4\) and \(n=8\) and compare to the exact value. Which integral provides a better algorithm for estimating \(\pi ?\)

For the functions \(f(x)=\left\\{\begin{array}{ll}2 x & \text { if } x<1 \\\ x^{2}+2 & \text { if } x \geq 1\end{array}\right.\) and \(g(x)=\left\\{\begin{array}{ll}2 x & \text { if } x \leq 1 \\ x^{2}+2 & \text { if } x>1\end{array}\right.\), assume that \(\int_{0}^{2} f(x) d x\) and \(\int_{0}^{2} g(x) d x\) exist. Explain why the approximating Riemann sums with midpoint evaluations are equal for any even value of \(n .\) Argue that this result implies that the two integrals are both equal to the sum \(\int_{0}^{1} 2 x d x+\int_{1}^{2}\left(x^{2}+2\right) d x\).

Find all functions satisfying the given conditions. $$f^{\prime \prime \prime}(x)=4-2 / x^{3}$$

Prove the following formula, which is basic to Simpson's Rule. If \(\quad f(x)=A x^{2}+B x+C, \quad\) then \(\int_{-h}^{h} f(x) d x=\frac{h}{3}[f(-h)+4 f(0)+f(h)]\)

Explain how you know the proposed integral value is wrong and (b) find all mistakes. $$\int_{0}^{\pi} \sec ^{2} x d x=\left.\tan x\right|_{x=0} ^{x=\pi}=\tan \pi-\tan 0=0$$

Find all functions satisfying the given conditions. $$f^{\prime \prime \prime}(x)=\sin x-e^{x}$$

Make the indicated substitution for an unspecified function \(f(x)\). $$u=\sin x \text { for } \int_{0}^{\pi / 2}(\cos x) f(\sin x) d x$$

A commonly used type of numerical integration algorithms is called Gaussian quadrature. For an integral on the interval \([-1,1],\) a simple Gaussian quadrature approximation is \(\int_{-1}^{1} f(x) d x \approx f\left(\frac{-1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right) .\) Show that, like Simpson's Rule, this Gaussian quadrature gives the exact value of the integrals of the power functions \(x, x^{2}\) and \(x^{3}\)

Find the position function \(s(t)\) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. $$v(t)=40-\sin t, s(0)=2$$

Compute \(\int_{0}^{4} f(x) d x\). $$f(x)=\left\\{\begin{array}{ll} 2 x & \text { if } x<1 \\ 4 & \text { if } x \geq 1 \end{array}\right.$$

Determine the position function if the velocity function is \(v(t)=3-12 t\) and the initial position is \(s(0)=3\)

Make the indicated substitution for an unspecified function \(f(x)\). $$u=\sqrt{x} \text { for } \int_{0}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x$$

Find the position function \(s(t)\) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. $$v(t)=40-\sin t, s(0)=2$$