미적분의 기본정리

The displacement from rest of a mass attached to a spring satisfies the simple harmonic motion equation $x(t)=A\cos (\omega t-\phi ),$ where $\phi$ is a phase constant, ω is the angular frequency, and A is the amplitude. Find the average velocity, the average speed (magnitude of velocity), the average displacement, and the average distance from rest (magnitude of displacement) of the mass.

If $f(x)>0,{f}^{'}(x)>0$ for all $x,$ then the right-hand rule underestimates the integral ${\int }_{a}^{b}f(x).$ Use a graph to justify your answer.

Chapter Review Exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

${\int }_{a}^{b}f{(x)}^{2}dx={\int }_{a}^{b}f(x)dx{\int }_{a}^{b}f(x)dx$

Chapter Review Exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

If $f(x)\le g(x)$ for all $x\in [a,b],$ then ${\int }_{a}^{b}f(x)\le {\int }_{a}^{b}g(x).$

Chapter Review Exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

All continuous functions have an antiderivative.

Evaluate the Riemann sums ${L}_{4}\,\text{and}\,{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

$y=3{x}^{2}-2x+1$ over $[-1,1]$

Evaluate the Riemann sums ${L}_{4}\,\text{and}\,{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

$y=\ln ({x}^{2}+1)$ over $[0,e]$

Evaluate the Riemann sums ${L}_{4}\,\text{and}\,{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

$y={x}^{2}\sin x$ over $[0,\pi ]$

Evaluate the Riemann sums ${L}_{4}\,\text{and}\,{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

$y=\sqrt{x}+\frac{1}{x}$ over $[1,4]$

Evaluate the following integrals.

${\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx$

Evaluate the following integrals.

${\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt$

Evaluate the following integrals.

${\int }_{\pi /3}^{\pi /2}2\sec (2\theta )\tan (2\theta )d\theta$

Evaluate the following integrals.

${\int }_{0}^{\pi /4}{e}^{{\cos }^{2}x}\sin x\cos xdx$

Find the antiderivative.

$\int \frac{dx}{{(x+4)}^{3}}$

Find the antiderivative.

$\int x\ln ({x}^{2})dx$

Find the antiderivative.

$\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx$

Find the antiderivative.

$\int \frac{{e}^{2x}}{1+{e}^{4x}}dx$

Find the derivative.

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

Find the derivative.

$\frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt$

Find the derivative.

$\frac{d}{dx}{\int }_{1}^{\ln (x)}(4t+{e}^{t})dt$