Exam 8: Integrals and Transcendental Functions

Determine if the given function y = f(x) is a solution of the accompanying differential equation. -Differential equation: $2 x y ^ { \prime } + 2 y = \cos x$ Initial condition: $y ( \pi ) = 0$ Solution candidate: $\mathrm { y } = \frac { 2 \sin \mathrm { x } } { \mathrm { x } }$

A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function. - $\sinh x = - \frac { 4 } { 3 } , \tanh x =$

Solve the problem. -A loaf of bread is removed from an oven at $350 ^ { \circ } \mathrm { F }$ and cooled in a room whose temperature is $70 ^ { \circ } \mathrm { F }$ . If the bread cools to $210 ^ { \circ } \mathrm { F }$ in 20 minutes, how much longer will it take the bread to cool to $185 ^ { \circ } \mathrm { F }$ .

Verify the integration formula. - $\int \tanh ^ { - 1 } x d x = x \tanh ^ { - 1 } x + \frac { 1 } { 2 } \ln \left( 1 - x ^ { 2 } \right) + C$

Verify the integration formula. - $\int 5 x \tanh x ^ { 2 } d x = \frac { 5 } { 2 } \ln \left( \cosh x ^ { 2 } \right) + C$

Find the slowest growing and the fastest growing functions as x→∞ . - y= y= y= y=

Find the slowest growing and the fastest growing functions as x→∞ . - y=2x y=5x y= y=

Solve the initial value problem. - $\frac { \mathrm { d } ^ { 2 } y } { d x ^ { 2 } } = 3 e ^ { - x } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$

Solve the problem. -A certain radioactive isotope decays at a rate of $1 \%$ per 400 years. If $t$ represents time in years and y represents the amount of the isotope left, use the condition that $\mathrm { y } = 0.99 \mathrm { y } 0$ to find the value of $\mathrm { k }$ in the equation $\mathrm { y } = \mathrm { y } _ { 0 } \mathrm { e } ^ { \mathrm { kt } }$ .

Solve the differential equation. - $\frac { d y } { d x } = 9 \sqrt { x y }$

Find the derivative of y with respect to the appropriate variable. - $\mathrm { y } = ( 4 - 4 \theta ) \tanh ^ { - 1 } \theta$

Evaluate the integral. - $\int 10 \sinh ( 4 x - \ln 2 ) d x$

Solve the problem. -The velocity of a body of mass $m$ falling from rest under the action of gravity is given by the equation $\mathrm { v } = \sqrt { \frac { \mathrm { mg } } { \mathrm { k } } } \tanh \left( \sqrt { \frac { \mathrm { gk } } { \mathrm { m } } \mathrm { t } } \right)$ , where $\mathrm { k }$ is a constant that depends on the body's aerodynamic properties and the density of the air, $g$ is the gravitational constant, and $t$ is the number of seconds into the fall. Find the limiting velocity, $\lim _ { \mathrm { t } \rightarrow \infty }$ , of a 210 lb. skydiver $( \mathrm { mg } = 210 )$ when $\mathrm { k } = 0.006$ .

Evaluate the integral. - $\int _ { 7 \pi / 6 } ^ { 7 \pi / 3 } 4 \cot \frac { t } { 7 } d t$

Rewrite the expression in terms of exponentials and simplify the results. - $6 \cosh ( \ln x )$

Evaluate the integral. - $\int x ^ { 3 } e ^ { - x ^ { 4 } } d x$

Verify the integration formula. - $\int 3 \operatorname { csch } 3 x d x = \ln \left| \tanh \frac { 3 } { 2 } x \right| + C$

Solve the problem. - $\text { Find the length of the curve } x=\frac{y^{2}}{32}-4 \ln \left(\frac{y}{5}\right), 8 \leq y \leq 16 \text {. }$

Evaluate the integral. - $\int _ { 1 } ^ { \mathrm { e } ^ { 5 } } \frac { 3 } { \mathrm { t } } \mathrm { dt }$

Evaluate the integral. - $\int _ { 0 } ^ { 5 \pi / 4 } \tan \frac { x } { 5 } d x$