SECTION 6.1 sectioN exercises 477
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
- x 1 2 3 4 f (x) 70 40 10 −20
- x 1 2 3 4 h(x) 70 49 34.3 24.01
- x 1 2 3 4 m (x) 80 61 42.9 25.61
- x 1 2 3 4 f (x) 10 20 40 80
- x 1 2 3 4 g (x) −3.25 2 7.25 12.5
For the following exercises, use the compound interest formula, A(t) = P 1 + r _ n nt.
- After a certain number of years, the value of an investment account is represented by the equation 10, 250 1 + 0.04 ____ 12
- What is the value of the account?
- What was the initial deposit made to the account in the previous exercise?
- How many years had the account from the previous exercise been accumulating interest?
- An account is opened with an initial deposit of $6,500 and earns 3.6% interest compounded semi-annually. What will the account be worth in 20 years?
- How much more would the account in the previous exercise have been worth if the interest were compounding weekly?
- Solve the compound interest formula for the principal, P.
- Use the formula found in Exercise #31 to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)
- How much more would the account in Exercises #31 and #34 be worth if it were earning interest for 5 more years?
- Use properties of rational exponents to solve the compound interest formula for the interest rate, r.
- Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.
- Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.
For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
- y = 3742(e)0.75t 40. y = 150 (e) 3.25 _ t 41. y = 2.25(e)−2t
- Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?
- How much less would the account from Exercise 42 be worth after 30 years if it were compounded monthly instead?
nUmeRIC For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
- f (x) = 2(5)x, for f (−3) 45. f (x) = −42x + 3, for f (−1) 46. f (x) = e x, for f (3)
- f (x) = −2e x − 1, for f (−1) 48. f (x) = 2.7(4)−x + 1 + 1.5, for f (−2) 49. f (x) = 1.2e2x − 0.3, for f (3)
- f (x) = − 3 _ 2 (3) −x + 3 _ 2 , for f (2)
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478 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs
TeChnOlOgy For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
- (0, 3) and (3, 375) 52. (3, 222.62) and (10, 77.456) 53. (20, 29.495) and (150, 730.89)
- (5, 2.909) and (13, 0.005) 55. (11,310.035) and (25,356.3652)
exTenSIOnS 56. The annual percentage yield (APY) of an investment
account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula APY = 1 + r __ 12
12 − 1.
- Repeat the previous exercise to find the formula for the APY of an account that compounds daily. Use the results from this and the previous exercise to develop a function I(n) for the APY of any account that compounds n times per year.
- Recall that an exponential function is any equation written in the form f (x) = a . b x such that a and b are positive numbers and b ≠ 1. Any positive number b can be written as b = en for some value of n. Use this fact to rewrite the formula for an exponential function that uses the number e as a base.
- In an exponential decay function, the base of the exponent is a value between 0 and 1. Thus, for some number b > 1, the exponential decay function can be written as f (x) = a . 1 _ b
x . Use this formula, along