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(a) Find the present value of $\$ 10,000$ per year flowing uniformly over a 7 year period if it earns $3 \%$ interest compounded continuously. (b) What is its final value?

(a) $\$ 63,138.58$(b) $\$ 77,892.69$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 8

Applications of the Definite Integral

Integrals

Baylor University

University of Nottingham

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

04:13

Find the future value and …

01:46

Determine the present valu…

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Find the present value of …

01:15

Present Value The present …

06:29

Suppose \$ 1000$ is invest…

Okay. Hello. So um in part a here um Well if we have p dollars that is deposited into an account paying an annual rate of interest um let's say are compounded end times per year. Then after two years the account will contain $8 according to this formula where A. Is equal to P. Times one Plus are over in to the tee times and power. So using this formula with P equal to $8906.54 t equal nine equal nine are equal to 3%. And and equal to two. We have that A. Is equal to 8000 900 six And 54 cents. Um Times one plus 0.0 3/2 raised to the two times 9 power. So that gives us that 8906.54 is going times or times one plus 0.15 So that's 1.0 15 to the nine times two. So to the 18th Um is going to be approximately equal to 11,000 uh 600 $43.88. So therefore this is the future value. Um The future value and then to find the interest earned. We just subtract the future value from the amount invested or in other words i is going to be equal to a minus P. So therefore that's just equal to 11,600 43 .88- the 8906 .54. Um which gives us the interest earned is going to be $2,737.34. Right? So these are interest earned. Um Now looking at part B. Um So again if P dollars is deposited at a rate of interest are compounded continuously for two years. Then be compounded amount $8 on the deposit is given by the formula. While A. Is equal to purchase. Well the amount we have is equal to the principal P. Times E. Raised to the R. T. So here again we have that we have that A. Is equal to our principal amounts of 8000 $906.54. Um Times E. Raised to the R. T. So that's 0.3 times nine. Which will be to the 0.27 And we compute this and we get this is approximately equal To $11,667 and 25 cents. So again there is our future value and to find the interest earned again i. Is equal to A minus P. So we have 11,667 and 25 cents minus the $8906.54 which gives us $2760.71. So therefore we have again. The interest earned is Uh $2,750.31.

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