적분 전체 문제

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] $y=x\sin ({x}^{2})$ over the interval $[-\pi ,0];$ the exact solution is $\frac{\cos ({\pi }^{2})-1}{2\pi }.$

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 $p(b)=\ell (b),$ and that ${\int }_{a}^{b}p(t)dt>{\int }_{a}^{b}\ell (t)dt.$ Explain why ${\int }_{c}^{d}p(t)>{\int }_{c}^{d}\ell (t)dt$ whenever $a\le c<d\le b.$

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

Suppose $[a,b]$ can be subdivided into subintervals $a={a}_{0}<{a}_{1}<{a}_{2}<\cdots <{a}_{N}=b$ such that either $f\ge 0$ over $[{a}_{i-1},{a}_{i}]$ or $f\le 0$ over $[{a}_{i-1},{a}_{i}].$ Set ${A}_{i}={\int }_{{a}_{i-1}}^{{a}_{i}}f(t)dt.$

a. Explain why ${\int }_{a}^{b}f(t)dt={A}_{1}+{A}_{2}+\cdots +{A}_{N}.$ b. Then, explain why $|{\int }_{a}^{b}f(t)dt|\le {\int }_{a}^{b}|f(t)|dt.$

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 $[{a}_{i-1},{a}_{i}]=1$ 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.

[T] Compute the left and right Riemann sums L₁₀ and R₁₀ and their average $\frac{{L}_{10}+{R}_{10}}{2}$ for $f(t)={t}^{2}$ over $[0,1].$ Given that ${\int }_{0}^{1}{t}^{2}dt=0.\overset{-}{33},$ to how many decimal places is $\frac{{L}_{10}+{R}_{10}}{2}$ accurate?

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?

If ${\int }_{1}^{5}\sqrt{1+{t}^{4}}dt=41.7133...,$ what is ${\int }_{1}^{5}\sqrt{1+{u}^{4}}du?$

Estimate ${\int }_{0}^{1}tdt$ 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 ${\int }_{0}^{1}tdt?$

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

If f is 1-periodic and ${\int }_{0}^{1}f(t)dt=A,$ is it necessarily true that ${\int }_{a}^{1+a}f(t)dt=A$ for all A?

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 $F(x)={\int }_{1}^{x}(1-t)dt.$ Find ${F}^{'}(2)$ and the average value of ${F}^{'}$ over $[1,2].$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{1}^{x}{e}^{-{t}^{2}}dt$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{1}^{x}{e}^{\cos t}dt$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{3}^{x}\sqrt{9-{y}^{2}}dy$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{3}^{x}\frac{ds}{\sqrt{16-{s}^{2}}}$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{x}^{2x}tdt$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{0}^{\sqrt{x}}tdt$

In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.

$\frac{d}{dx}{\int }_{0}^{\sin x}\sqrt{1-{t}^{2}}dt$