Compute the magnitude of the fundamental cross product for the surface defined parametrically.

$$\mathbf{R}(u, v)=u \mathbf{i}+v^{2} \mathbf{j}+u^{3} \mathbf{k}$$

(a) integrate to find F as a function of x, and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $F(x)=\int_{4}^{x} t^{3 / 2} d t$

Let $f(t)=F'(t).$ Write the integral $\int_{a}^{b}f(t)\ dt$ and evaluate it using the Fundamental Theorem of Calculus.

$$F(t) = t^{4}, a = −1, b = 1$$

A $12.5 \mathrm{~g}$ clothesline is stretched with a tension of $22.1 \mathrm{~N}$ between two poles $7.66 \mathrm{~m}$ apart. What is the frequency of the fundamental harmonic?

Use Part I of the Fundamental Theorem to compute each integral exactly. $\int_{\pi / 4}^{\pi / 2} 3 \csc x \cot x d x$

Do you think the sign of the charge of an elementary particle, like an electron or proton, is a more, or less fundamental property than the "sign" of its symmetry?

factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$\newline\sec^4x-\tan^4 x$$

Explain the significance of: Pilgrims, Mayflower Compact, Puritans, Great Migration, Fundamental Orders of Connecticut, Roger Williams, Anne Hutchinson, King Philip's War.

The given differential equation has a fundamental set of solutions whose Wronskian $W(t)$ is such that $W(0)=1$. What is $W(4)$ ?

$$y^{(4)}-y^{\prime \prime}+y=0$$

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $\csc \phi \tan \phi+\sec \phi$

Estimate the fundamental frequency of vibration of a telephone line between adjacent poles near where you live.Explain how you arrived at your answer.

factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer.

$$\newline\sin^4x-\cos^4 x$$

Use an organized list, a tree diagram, or the fundamental counting principle to find the number of permutations. the number of ways 6 shirts can be arranged in a closet

When a guitar string is vibrating in its fundamental mode, is the wavelength of the sound it produces in air typically the same as the wavelength of the standing wave on the string? Explain.