Superposition Principle. Let $y_1$ be a solution to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g_1(t)$$

on the interval $I$ and let $y_2$ be a solution to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g_2(t)$$

on the same interval. Show that for any constants $k_1$ and $k_2$, the function $k_1 y_1+k_2 y_2$ is a solution on $I$ to

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=k_1 g_1(t)+k_2 g_2(t) \text {. }$$

Determine the following limits. a. $\lim _{t \rightarrow-2^{+}} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. b. $\lim _{t \rightarrow-2^{-}} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. c. $\lim _{t \rightarrow-2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$. d. $\lim _{t \rightarrow 2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$.

Prove: If x=x(t) and y=y(t) are differentiable at t, and if z=f(x, y) is differentiable at the point (x(t), y(t)), then

$$\frac { d z } { d t } = \nabla z \cdot \mathbf { r } ^ { \prime } ( t )$$

where r(t)=x(t)i+y(t)j.

Read about the disjunctive normal form of a statement in a logic text.

a. What is the disjunctive normal form of a statement that has the following truth table?

Explain why the disjunctive normal form is a valuable concept.

If $F(t)=t(\ln t)-t$, find $F^{\prime}(t)$.

Let $V$ be the inner product space in previous example. (a) If $p(t)=\sqrt{t}$, find $q(t)=a+b t \neq 0$ such that $p(t)$ and $q(t)$ are orthogonal. (b) If $p(t)=\sin t$, find $q(t)=a+b e^{\prime} \neq 0$ such that $p(t)$ and $q(t)$ are orthogonal.

Consider $V$ to be the inner product space of Example 6. Find the cosine of the angle between each pair of vectors in $V$.

(a) $p(t)=t, q(t)=t-1$ (b) $p(t)=t, q(t)=t$ (c) $p(t)=1, q(t)=2 t+3$

In the following exercise, find T(t), N(t), a$_{\mathrm{T}}$, and a$_{\mathrm{N}}$ at the given time t for the plane curve r(t).

$$\mathbf{r}(t)=\langle\cos \omega t+\omega t \sin \omega t, \sin \omega t-\omega t \cos \omega t\rangle, \quad t=t_0$$

$x(t)=t^{1 / 3}, y(t)=t^{2}$ from t=1 to t=1.1.

Find the convolution $f ( t ) * g ( t )$.

$$f ( t ) = t ^ { 2 } , g ( t ) = \cos t$$

Let the moment-generating function M(t) of X exist for −h < t < h. Consider the function R(t) = ln M(t). The first two derivatives of R(t) are, respectively, R'(t) = M'(t)/M(t) and R"(t) = M(t) M"(t) - [M'(t)]²/[M(t)]². Setting t = 0, show that a. μ = R'(0). b. σ² = R"(0).

Suppose $f(t)$ is a periodic function with period $T$; that is,

$$f(t+T)=f(t) \text { for all } t .$$

Show that

$$\mathscr{L}[f]=\frac{1}{1-e^{-T s}} \int_0^T f(t) e^{-s t} d t$$

Complete the truth table.

Evaluate the following integrals $(r \neq 0)$ :

(a) $\int_0^T b t e^{-r t} d t$

(b) $\int_0^T(a+b t) e^{-r t} d t$

(c) $\int_0^T\left(a-b t+c t^2\right) e^{-r t} d t$