In each of these cases, find the rate of change of f(t) with respect to t at the given value of t. a. $f(t)=t^{3}-4 t^{2}+5 t \sqrt{t}-5$ at t = 4 b. $f(t)=\frac{2 t^{2}-5}{1-3 t}$ at t = -1

In each of these cases, find the rate of change of $f(t)$ with respect to $t$ at the given value of $t$.

a. $f(t)=t^3-4 t^2+5 t \sqrt{t}-5$ at $t=4$

b. $f(t)=\frac{2 t^2-5}{1-3 t}$ at $t=-1

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.

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

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

Prove the property. In each case, assume r, u, and v are differentiable vector-valued functions of t in space, w is a differentiable real-valued function of t, and c is a scalar. d/dt{r(t) · [u(t) × v(t)} = r′(t) · [u(t) × v(t)] + r(t) · [u′(t) × v(t)] + r(t) · [u(t) × v′(t)]

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 Laplace transforms of these functions: (a) $f(t)=(t-4) u(t-2)$ (b) $g(t)=2 e^{-4 t} u(t-1)$ (c) $h(t)=5 \cos (2 t-1) u(t)$ (d) $p(t)=6[u(t-2)-u(t-4)]$

Which of the following are odd functions? (a) sin t + cos t (b) t sin t (c) t ln t (d) t³ cos t (e) sinh t

Find $\mathbf{r}^{\prime \prime}(t)$.

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

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

Write the given system in the matrix form x' = Ax + f.

$$x'(t)=3x(t)-y(t)+t^2$$

$$y'(t)=-x(t)+2y(t)+e^t$$

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

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

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

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

Show that

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