1 / 9

4.5

4.5. THE NUMBER e. The Natural Number e.

bevis-solomon
Download Presentation

4.5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4.5 THE NUMBER e

  2. The Natural Number e An irrational number, introduced by Euler in 1727, is so important that it is given a special name, e. Its value is approximately e ≈ 2.71828 . . .. It is often used for the base, b, of the exponential function. Base e is called the natural base. This may seem mysterious, as what could possibly be natural about using an irrational base such as e? The answer is that the formulas of calculus are much simpler if e is used as the base for exponentials.

  3. Exponential Functions with Base e For the exponential function Q = a bt, the continuous growth rate, k, is given by solving ek = b. Then Q = a ekt. If a is positive, • If k > 0, then Q is increasing. • If k < 0, then Q is decreasing.

  4. Exponential Functions with Base e Example 1 Give the continuous growth rate of each of the following functions and graph each function: P = 5e0.2t, Q = 5e0.3t, and R = 5e−0.2t. Solution The function P = 5e0.2t has a continuous growth rate of 20%, Q = 5e0.3t has a continuous 30% growth rate, and R = 5e−0.2t has a continuous growth rate of −20%. The negative sign in the exponent tells us that R is decreasing instead of increasing. Because a = 5 in all three functions, they each pass through the point (0,5). They are all concave up and have horizontal asymptote y = 0. Q = 5e0.3t P = 5e0.2t R = 5e−0.2t

  5. Exponential Functions with Base e Example 3 Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? Solution If A is the amount of caffeine in the body t hours after drinking the coffee, then A = 100e−0.17t. Note that the continuous growth rate is −17% since A is decreasing. After 8 hours, we have A = 100e−0.17(8) = 25.67 mg.

  6. If , with t in years, we say that P is growing at an annual rate of 7%. If , with t in years, we say that P is growing at an continuous rate of 7%. Since we can rewrite . In other words a 7% continuous rate and a 7.25% annual rate generate the same increases in P.

  7. Connection: The Number eand Compound Interest Interesting Tables Balance after 1 year, 100% nominal interest various compounding frequencies. Compare last number to e. Effect of increasing the frequency of compounding, 6% nominal interest.

  8. Connection: The Number e and Compound Interest If interest on an initial deposit of $P is compounded continuously at a nominal rate of r per year, the balance t years later can be calculated using the formula B = P ert. For example, if the nominal rate is 6%, then r = 0.06.

  9. Exponential Functions with Base e Example 4 In November 2005, the Wells Fargo Bank offered interest at a 2.323% continuous yearly rate. Find the effective annual rate. Solution Since e0.02323 = 1.0235, the effective annual rate is 2.35%. As expected, the effective annual rate is larger than the continuous yearly rate.

More Related