Let

$$A=\left[\begin{array}{ll} 2 & -5 \\ 2 & -3 \end{array}\right]$$

Without explicitly computing the eigenvalues of A, decide whether the real parts of both eigenvalues are negative.

Sketch the region between y = x + 4 and y = 2 - x for

$$0 \leq x \leq 2.$$

Using symmetry, explain why the centroid of the region lies on the line y = 3. Verify this by computing the moments and the centroid.

Verify Stokes’ Theorem by computing both integrals. S is the portion of $z=\sqrt{1-x^{2}-y^{2}}$ above the xy-plane, $\mathbf{F}=\left\langle 2 x, z^{2}-x, x z^{2}\right\rangle$.

Let f be the function given by $f(x)=\sin \left(5 x+\frac{\pi}{4}\right)$, and let $P_{3}(x)$ be the third-degree Taylor polynomial for f about x=0. Find P(x) by computing the coefficients.

The measurement of the edge of a cube is found to be 12 inches, with a possible error of $0.03$ inch. Use differentials to approximate the maximum possible propagated error in computing (a) the volume of the cube and (b) the surface area of the cube.

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) $\int \ln x d x$

The measurement of the circumference of a circle is found to be 60 centimeters, with a possible error of $1.2$ centimeters.

Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed $3\%$.

Let A and B be invertible matrices. (a) By computing an appropriate matrix product, verify that

$$\left( A ^ { - 1 } B \right) ^ { - 1 } = B ^ { - 1 } A.$$

(b) Use properties of the inverse to derive

$$\left( A ^ { - 1 } B \right) ^ { - 1 } = B ^ { - 1 } A.$$

Area The measurement of the side of a square floor tile is $10$ inches, with a possible error of $\frac{1}{32}$ inch. Use differentials to approximate the possible propagated error in computing the area of the square.

Let

$$\mathcal { R } = [ 0 , 1 ] \times [ 0 , 1 ].$$

Estimate

$$\int \int _ { \mathcal { R } } ( x + y ) d A$$

by computing two different Riemann sums, each with at least six rectangles.

A function f is given along with a number c in its domain. (a.) Find the difference quotient of f. (b.) Find $f^{\prime}(c)$ by computing the limit of the difference quotient.

$$f(x)=2-x^{2} \text { at } c=0$$

Let

$$A=\left[\begin{array}{rr} 0 & 1 \\ 2 & -1 \end{array}\right]$$

Without explicitly computing the eigenvalues of A, decide whether the real parts of both eigenvalues are negative.

The measurement of the radius of the end of a log is found to be $14$ inches, with a possible error of $\frac{1}{4}$ inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

Describe your sample space, your assignment of probabilities, and the event whose probability you are computing. Two dice are thrown eight times. Compute the probability that the sum is 11 exactly three times.