Investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a. $f ( x ) = ( x - a ) ^ { 2 }$

Let X have a Poisson distribution with parameter

$$\lambda$$

. Use the expansion

$$e ^ { t / \sqrt { \lambda } } = \sum _ { i = 0 } ^ { \infty } \frac { [ t / \sqrt { \lambda } ] ^ { i } } { i ! }$$

to show that

$$\lim _ { \lambda \rightarrow \infty } m _ { Y } ( t ) = e ^ { t ^ { 2 } / 2 }$$

.

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. If an interval fo is not specified, assume that

$$- \infty < t < \infty$$

$$x = t , y = - 2 t$$

The position of an object at time t is given. (a) Eliminate the parameter t to find y as a function of x. (b) Graph $\mathbf{r}=\mathbf{r}(t)$ and indicate the orientation. $\mathbf{r}(t)=\left(1-t^{2}\right) \mathbf{i}+t^{2} \mathbf{j}$

If C is the graph of a smooth vector-valued function r(t), then the unit binormal vector to C at t may be computed directly in terms of r'(t) and r''(t) by the formula B(t)=. When t=s is the arc length parameter, this formula simplifies to B(s)=.

For an x-ray diffraction pattern (having all peaks plane-indexed) of a metal that has a unit cell of cubic symmetry, generate a spreadsheet that allows the user to input the x-ray wavelength, and then determine, for each plane, the following: (a) $\mathrm{d}_{h k l}$ (b) The lattice parameter, a

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval fo is not specified, assume that

$$- \infty < t < \infty$$

$$x = t , y = 2 t$$

Evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. F=$\langle y , - x \rangle$; C is the circle of radius 3 centered at the origin oriented counterclockwise.

Is the statement below true or false? Give reasons for your answer.

The line parameterized by $x=1, y=2 t, z=3+t$ is parallel to the x-axis.

Plot the following curves, indicating the positive orientation.

$x=e ^ { t } , y = 3 e ^ { - 2 t }$, for $- \infty < t < \infty: (1, 3)$

Investigate the one-parameter family of functions. Assume that a is positive. (a) Graph f(x) using three different values for a. (b) Using your graph in part (a), describe the critical points of f and how they appear to move as a increases. (c) Find a formula for the x-coordinates of the critical point(s) of f in terms of a. $f ( x ) = x ^ { 3 } - a x$

Assume that the binomial parameter p is to be estimated with the function $\frac{X}{n},$ where X is the number of successes in n independent trials. Which demands the larger sample size: requiring that $\frac{X}{n}$ have a 96% probability of being within 0.05 of p, or requiring that $\frac{X}{n}$ have a 92% probability of being within 0.04 of p?

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.

$$y ^ { \prime \prime } - 7 y ^ { \prime } + 10 y = 24 e ^ { x }$$

$$y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 5 x } + 6 e ^ { x } , ( - \infty , \infty )$$

sketch the images of the following paths, using arrows to indicate the direction in which the parameter increases:

$$\left\{ \begin{array} { l } { x = t \cos t } \\ { y = t \sin t } \end{array} \right. \quad - 6 \pi \leq t \leq 6 \pi$$