In the following exercise, find T(t), N(t), a$_{\mathrm{T}}$, and a$_{\mathrm{N}}$ at the given time t for the space curve r(t). [Hint: Find a(t), T(t), and a$_{\mathrm{N}}$. Solve for N in the equation a(t)=a$_{\mathrm{T}}$T + a$_{\mathrm{N}}$N.]

Function: $\mathbf{r}(t)=e^t \sin t \mathbf{i}+e^t \cos t \mathbf{j}+e^t \mathbf{k}$, Time: $t=0$

Find T(t) and N(t) at the given point. x=cosh t, y=sinh t, z=t, t=ln 2

Let

$$f ( t ) = 10 t - t ^ { 2 }$$

and

$$g ( t ) = \frac { t ^ { 4 } - 10 t ^ { 3 } - 2 t ^ { 2 } + 20 t } { t ^ { 6 } }$$

. Find

$$f ( t ) \div g ( t )$$

.

Find the convolution $f(t)*g(t)$ for the given problems:

$$a)\; f(t)=t,\; g(t)=1,\; b)\; f(t)=g(t)=\sin t,\; c)\; f(t)=g(t)=e^{at}.$$

In the following exercise, find T(t), N(t), a$_{\mathrm{T}}$, and a$_{\mathrm{N}}$ at the given time t for the space curve r(t). [Hint: Find a(t), T(t), and a$_{\mathrm{N}}$. Solve for N in the equation a(t)=a$_{\mathrm{T}}$T + a$_{\mathrm{N}}$N.]

Function: $\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}-3 t \mathbf{k}$, Time: $t=1$

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

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

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

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

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. $\lim _{t \rightarrow 2} \frac{t^{3}-5 t^{2}+6 t}{t^{4}-4 t^{2}}$.

Write the product using an exponent. t • t • t • t • t • t

If A and B are differentiable functions of t, find the derivative of the function with respect to t. Give your result in terms of $A(t), A^{\prime}(t), B(t)$, and $B^{\prime}(t)$. (a) $y=A(t) B(t)$ (b) $y=[A(t)]^2+[B(t)]^2$ (c) $y=[A(t)]^2 B(t)$

r(t) = ⟨e^-t, t², tan t⟩ Find (a)r′(t), (b)r″(t), (c)r′(t)·r″(t) and (d)r′(t)×r″(t).

Which of the following sets of vectors are bases for $P_3$ ?

(a) $\left\{t^3+2 t^2+3 t, 2 t^3+1,6 t^3+8 t^2+6 t+4\right.$, $\left.t^3+2 t^2+t+1\right\}$

(b) $\left\{t^3+t^2+1, t^3-1, t^3+t^2+t\right\}$

(c) $\left\{t^3+t^2+t+1, t^3+2 t^2+t+3,2 t^3+t^2+3 t+2\right.$, $\left.t^3+t^2+2 t+2\right\}$

(d) $\left\{t^3-t, t^3+t^2+1, t-1\right\}$

Find F(s) if: (a) $f(t)=6 e^{-t} \cosh 2 t$ (b) $f(t)=3 t e^{-2 t} \sinh 4 t$ (c) $f(t)=8 e^{-3 t} \cosh t u(t-2)$

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)]