Exam 14: Further Integration Techniques and Applications of the Integral

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 x y$ Solve for y as a function of x.

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { 1 } \frac { 1 } { x ^ { \frac { 1 } { 7 } } } d x$

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( x + 3 ) y ^ { 2 } }$ Solve for y as a function of x.

The U. S. Treasury issued a 20-year bond on November 16, 1998, paying 6.07% interest. Thus, if you bought $100,000 worth of these bonds, you would receive $6,070 per year in interest for 20 years. At investor wishes to buy the rights to receive the interest on $100,000 worth of these bonds. The amount the investor is willing to pay is the present value of the interest payments, assuming a 6% rate of return. If we assume (incorrectly, but approximately) that the interest payments are made continuously, what will the investor pay Round your answer to the nearest cent.

Evaluate the integral. $\int \left( x ^ { 2 } - 8 \right) e ^ { 5 x } \mathrm {~d} x$

The annual net sales (revenue) earned by the Finnish cell phone company Nokia from January 1999 to January 2004 can be approximated by $R ( t ) = - 1.7 t ^ { 2 } + 5 t + 28$ billion euros per year $( - 1 \leq t \leq 4 )$ where t is time in years ( $t = 0$ represents January 2000). Suppose that, from January 1999 on, Nokia invested its revenue in an investment yielding 7% compounded continuously. What, to the nearest €10 billion, would the total value of Nokia's revenues have been at the end of 2003

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. ​ $R ( t ) = 80,000$ , $0 \leq t \leq 20$ , at 11% ​ Please enter your answer in the form TV = , FV = . Round FV to the nearest cent. ​ TV = $__________ FV = $__________

A rocket rising from the ground has a velocity of $\text { 1,700te }^{-\frac{t}{100}}$ $\frac { \mathrm { ft } } { \mathrm { s } }$ after t seconds. How far does it rise in the first 1 minutes and 40 seconds

Find the formula for the c-unit moving average of a general linear function. $f ( x ) = n x + q$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 4 } ^ { + \infty } e ^ { - 0.25 x } \mathrm {~d} x$

Your monthly salary has been increasing steadily for the past year, and your average monthly salary over the past year was x dollars. Would you have earned more money if you had been paid ?x dollars per month ?

The Laplace transform F(x) of a function f (t ) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$ Find $F ( x )$ if $f ( t ) = 8$ .

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$ , $\pi = 3.14159265 \ldots$ and and are constants called the standard deviation and the mean, respectively. With $\sigma = 3$ and $\mu = 5$ , approximate $\int _ { - \infty } ^ { \infty } p ( x ) \mathrm { d } x$ .

Your monthly sales of Tofu Ice Cream are falling at an instantaneous rate of 3% per month. If you currently sell $s = 1,070$ per month, solve the differential equation that describes your change in sales to predict your monthly sales. Assume that the current sales are given for time $t = 0$ . NOTE: Enter your answer in the form $s = f ( t )$ , where f is some function of t.

Find the total value or the given income stream and also find its future value (at the end of the given interval) using the given interest rate. $R ( t ) = 50,000 e ^ { 0.04 t }$ , $0 \leq t \leq 5$ , at 7%

Some values of a function and its 3-unit moving average are given. Supply the missing information: 1 2 3 4 5 6 7 ( ) 2 3 14 4 20 average undefined undefined 5 11

You are saving for your retirement by investing $500 per month in an annuity with a guaranteed interest rate of 6% per year. You increase the amount you invest at the rate of 3% per year. With continuous investment and compounding, how much will you have accumulated in the annuity by the time you retire in 35 years Round your answer to the nearest cent.

Find the total value or the given income stream and also find its future value (at the end of the given interval) using the given interest rate. ​ $R ( t ) = 40,000 e ^ { 0.04 t }$ , $0 \leq t \leq 20$ , at 7% ​ Please enter your answer in the form TV = , FV = . Give the answers to the nearest cent. ​ TV = $__________ FV = $__________