Exam 14: Further Integration Techniques and Applications of the Integral

For the given differential equation, find the particular solution if $y ( 3 ) = 12$ . $x \frac { \mathrm { d } y } { \mathrm {~d} x } = y$

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 80,000 + 2,000 t$ , $0 \leq t \leq 10$ , at 8%

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. ​ $R ( t ) = 20,000 + 3,000 t$ , $0 \leq t \leq 15$ , at 10% ​ Please enter your answer in the form TV = , FV = . Round FV to the nearest cent. ​ TV = $__________ FV = $__________

Evaluate the integral. $\int \frac { 4 x + 3 } { e ^ { 7 x } } \mathrm {~d} x$

Your monthly profit p on sales of Avocado Ice Cream are rising at an instantaneous rate of 17% per month. If you currently make a profit of $14,500 per month, solve the differential equation that describes your change in profit to predict your monthly profits. Assume that the current sales are given for time $t = 0$ .

If the integral converges, compute its value. $\int _ { - \infty } ^ { 2 } e ^ { x } \mathrm {~d} x$

If you invest $20,000 at 7% interest compounded continuously, what is the average amount in your account over 1 year Please round your answer to the nearest cent.

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. ​ $R ( t ) = 90,000 e ^ { 0.07 t }$ , $0 \leq t \leq 25$ , at 10% ​ Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary. ​ TV = $__________ PV = $__________

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x$

Evaluate the integral. $\int 3 x e ^ { x } \mathrm {~d} x$

Evaluate the integral. $\int - 7 x e ^ { - x } \mathrm {~d} x$

For the differential equation, find the particular solution if $y ( 0 ) = 1$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ( y + 5 )$

Find the average of the function $f ( x ) = 2 x ^ { 3 }$ over $[ 0,8 ]$ .

Find the area of the indicated region. Enclosed by $y = \ln x$ and $y = 2 x - 3$ . (Round answer to four significant digits.) [First use technology to determine approximately where the graphs cross.]

The annual net sales (revenue) earned by the Finnish cell phone company Nokia from January 1999 to January 2003 can be approximated by $R ( t ) = - 1.7 t ^ { 2 } + 5 t + 28$ billion euros per year $( - 1 \leq t \leq 3 )$ where t is time in years ( $t = 0$ represents January 2000). Estimate, to the nearest billion, Nokia's total revenue from January 1999 through December 2002.

For the differential equation, find the particular solution if $y \left( \frac { 1 } { 7 } \right) = e ^ { 7 }$ $x ^ { 2 } \frac { \mathrm { d } y } { \mathrm {~d} x } = y$

Find the total value of the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 70,000$ , $0 \leq t \leq 15$ , at 6%

Luckily, sales of your Star Wars T-shirts are now 57 T-shirts per week and increasing continuously at a rate of 3% per week. You are now charging $14 per T-shirt and are decreasing the price by 30 cents per week. How much revenue will you generate during the next six weeks Round your answer to the nearest cent.

Find the average of the function $f ( x ) = e ^ { x }$ over $[ 0,8 ]$ .

When your first child is born, you begin to save for college by depositing $500 per month in an account paying 12% interest per year. You increase the amount you save by 2% per year. With continuous investment and compounding, how much will you have accumulated in the account by the time your child enters college 18 years later Round your answer to the nearest cent.