Chapter 5

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=\sin x, a=0, b=2 \pi\)

In the following exercises, find the average value \(f_{\text { ave }}\) of \(f\) between \(a\) and \(b,\) and find a point \(c,\) where \(f(c)=f_{\text { ave. }}\) \(f(x)=\cos x, a=0, b=2 \pi\)

In the following exercises, approximate the average value using Riemann sums \(L_{100}\) and \(R_{100} .\) How does your answer compare with the exact given answer? [T] \(y=\ln (x)\) over the interval \([1,4] ;\) the exact solution is \(\frac{\ln (256)}{3}-1\)

In the following exercises, approximate the average value using Riemann sums \(L_{100}\) and \(R_{100} .\) How does your answer compare with the exact given answer? [T] \(y=e^{x / 2}\) over the interval \([0,1] ;\) the exact solution is 2\((\sqrt{e}-1)\)

In the following exercises, approximate the average value using Riemann sums \(L_{100}\) and \(R_{100} .\) How does your answer compare with the exact given answer? [T] \(y=\tan x\) over the interval \(\left[0, \frac{\pi}{4}\right] ;\) the exact solution is \(\frac{2 \ln (2)}{\pi} .\)

In the following exercises, approximate the average value using Riemann sums \(L_{100}\) and \(R_{100} .\) How does your answer compare with the exact given answer? [T] \(y=\frac{x+1}{\sqrt{4-x^{2}}}\) over the interval \([-1,1] ;\) the exact solution is \(\frac{\pi}{6}\)

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? [T] \(y=x^{2}-4\) over the interval \([0,2] ;\) the exact solution is \(-\frac{8}{3}\)

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? \([\mathrm{T}] \quad y=x e^{x^{2}}\) over the interval \([0,2] ;\) the exact solution is \(\frac{1}{4}\left(e^{4}-1\right)\)

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? \([\mathrm{T}] \quad y=\left(\frac{1}{2}\right)^{x}\) over the interval \([0,4] ;\) the exactsolution is \(\frac{15}{64 \ln (2)}\)

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? [T] \(y=x \sin \left(x^{2}\right)\) over the interval \([-\pi, 0] ;\) the exact solution is \(\frac{\cos \left(\pi^{2}\right)-1}{2 \pi}\)

Suppose that \(A=\int_{0}^{2 \pi} \sin ^{2} t d t\) and \(B=\int_{0}^{2 \pi} \cos ^{2} t d t\). Show that \(A+B=2 \pi\) and \(A=B\).

In the following exercises, compute the average value using the left Riemann sums \(L_{N}\) for \(N=1,10,100\) . How does the accuracy compare with the given exact value? Suppose that \( A=\int_{-\pi / 4}^{\pi / 4} \sec ^{2} t d t=\pi \qquad\) and \(B=\int_{-\pi / 4}^{\pi / 4} \tan ^{2} t d t .\) Show that \(A-B=\frac{\pi}{2}\).

Suppose that \(A=\int_{-\pi / 4}^{\pi / 4} \sec ^{2} t d t=\pi\) and \(B=\int_{-\pi / 4}^{\pi / 4} \tan ^{2} t d t\). Show that \(A-B=\frac{\pi}{2}\)

Show that the average value of \(\sin ^{2} t\) over \([0,2 \pi]\) is equal to 1\(/ 2\) Without further calculation, determine whether the average value of \(\sin ^{2} t\) over \([0, \pi]\) is also equal to 1\(/\)2.

Show that the average value of \(\cos ^{2} t\) over \([0,2 \pi]\) is equal to 1\(/ 2\) . Without further calculation, determine whether the average value of \(\cos ^{2}(t)\) over \([0, \pi]\) is also equal to 1\(/ 2\)

Explain why the graphs of a quadratic function (parabola) \(p(x)\) and a linear function \(\ell(x)\) can intersect in at most two points. Suppose that \(p(a)=\ell(a)\) and \(p(b)=\ell(b), \quad\) and that \(\int_{a}^{b} p(t) d t>\int_{a}^{b} \ell(t) d t .\) Explain why \(\int_{c}^{d} p(t)>\int_{c}^{d} \ell(t) d t\) whenever \(a \leq c< d \leq b\)

Suppose that parabola \(p(x)=a x^{2}+b x+c\) opens downward \((a<0)\) and has a vertex of \(y=\frac{-b}{2 a}>0 .\) For which interval \([A, B]\) is \(\int_{A}^{B}\left(a x^{2}+b x+c\right) d x\) as large as possible?

Suppose \([a, b]\) can be subdivided into subintervals \(a=a_{0}

Suppose \(f\) and \(g\) are continuous functions such that \(\int_{c}^{d} f(t) d t \leq \int_{c}^{d} g(t) d t\) for every subinterval \([c, d]\) of \([a, b] .\) Explain why \(f(x) \leq g(x)\) for all values of \(x .\)

Suppose that \([a, b]\) can be partitioned. taking \(a=a_{0}< a_{1}< \cdots< a_{N}=b\) such that the average value of \(f\) over each subinterval \(\left[a_{i-1}, a_{i}\right]=1\) is equal to 1 for each \(i=1, \ldots, N .\) Explain why the average value of \(f\) over \([a, b]\) is also equal to \(1 .\)

Suppose that \([a, b]\) can be partitioned. taking \(a=a_{0}

Suppose that for each \(i\) such that \(1 \leq i \leq N\) one has \(\int_{i-1}^{i} f(t) d t=i .\) Show that \(\int_{0}^{N} f(t) d t=\frac{N(N+1)}{2}\)

Suppose that for each \(i\) such that \(1 \leq i \leq N\) one has \(\int_{i-1}^{i} f(t) d t=i^{2}\) show that \(\int_{0}^{N} f(t) d t=\frac{N(N+1)(2 N+1)}{6}\).

[T] Compute the left and right Riemann sums \(L_{10}\) and \(R_{10}\) and their average \(\frac{L_{10}+R_{10}}{2}\) for \(f(t)=t^{2}\) over \([0,1] .\) Given that \(\int_{0}^{1} t^{2} d t=0 . \overline{33}, \quad\) to how many decimal places is \(\frac{L_{10}+R_{10}}{2}\) accurate?

[T] Compute the left and right Riemann sums, \(L_{10}\) and \(R_{10},\) and their average \(\frac{L_{10}+R_{10}}{2}\) for \(f(t)=\left(4-t^{2}\right)\) over \([1,2] .\) Given that \(\int_{1}^{2}\left(4-t^{2}\right) d t=1 . \overline{66},\) to how many decimal places is \(\frac{L_{10}+R_{10}}{2}\) accurate?

If \( \int_{1}^{5} \sqrt{1+t^{4}} d t=41.7133 \ldots, \) what is \(\int_{1}^{5} \sqrt{1+u^{4}} d u ?\)

If \(\int_{1}^{5} \sqrt{1+t^{4}} d t=41.7133 \ldots\).. what is \(\int_{1}^{5} \sqrt{1+u^{4}} d u ?\)

Estimate \(\int_{0}^{1} t d t\) using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value \(\int_{0}^{1} t d t ?\)

Estimate \(\int_{0}^{1} t d t\) by comparison with the area of a single rectangle with height equal to the value of \(t\) at the midpoint \(t=\frac{1}{2} .\) How does this midpoint estimate compare with the actual value \(\int_{0}^{1} t d t ?\)

If \(f\) is 1 -periodic \((f(t+1)=f(t)), \quad\) odd, and integrable over \([0,1], \quad\) is it always true that \(\int_{0}^{1} f(t) d t=0 ?\)

If \(f\) is 1-periodic \((f(t+1)=f(t)),\) odd, and integrable over \([0,1],\) is it always true that \(\int_{0}^{1} f(t) d t=0 ?\)

If \(f\) is 1 -periodic and \(\int_{0}^{1} f(t) d t=A, \quad\) is it necessarily true that \(\int_{a}^{1+a} f(t) d t=A\) for all \(A\) ?

Consider two athletes running at variable speeds \(v_{1}(t)\) and \(v_{2}(t) .\) The runners start and finish a race at exactly the same time. Explain why the two runners must be going the same speed at some point.

Two mountain climbers start their climb at base camp, taking two different routes, one steeper than the other, and arrive at the peak at exactly the same time. Is it necessarily true that, at some point, both climbers increased in altitude at the same rate?

To get on a certain toll road a driver has to take a card that lists the mile entrance point. The card also has a timestamp. When going to pay the toll at the exit, the driver is surprised to receive a speeding ticket along with the toll. Explain how this can happen.

Set \(\quad F(x)=\int_{1}^{x}(1-t) d t .\) Find \(\quad F^{\prime}(2)\) and the average value of \(F^{\prime}\) over \([1,2]\)

Set \(F(x)=\int_{1}^{x}(1-t) d t\). Find \(F^{\prime}(2)\) and the average value of \(F^{\prime}\) over [1,2] .

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{1}^{x} e^{-t^{2}} d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{1}^{x} e^{\cos t} d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{3}^{x} \sqrt{9-y^{2}} d y $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{4}^{x} \frac{d s}{\sqrt{16-s^{2}}} $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{x}^{2 x} t d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1 , to find each derivative. \(\frac{d}{d x} \int_{0}^{\sqrt{x}} t d t\)

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{0}^{\sin x} \sqrt{1-t^{2}} d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1 , to find each derivative. \(\frac{d}{d x} \int_{\cos x}^{1} \sqrt{1-t^{2}} d t\)

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{1}^{\sqrt{x}} \frac{t^{2}}{1+t^{4}} d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{1}^{x^{2}} \frac{\sqrt{t}}{1+t} d t $$

In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{0}^{\ln x} e^{t} d t $$

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ y=x^{2} \text { over }[0,4] $$

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ y=x^{3}+6 x^{2}+x-5 \text { over }[-4,2] $$