(10 ) WARM UP Simplify the expression. 1. e –4x e12x ANSWER

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Presentation transcript:

(10 ) WARM UP Simplify the expression. 1. e –4x 3 1000 e12x ANSWER Use a calculator to evaluate 2. e 3/4 . about 2.117 ANSWER

EXAMPLE 1 Graph natural base functions Graph the function. State the domain and range. y = 3e 0.25x SOLUTION Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plot the points (0, 3) and (1, 3.85) and draw the curve. The domain is all real numbers, and the range is y > 0.

EXAMPLE 2 Graph natural base functions Graph the function. State the domain and range. y = e –0.75(x – 2) + 1 SOLUTION a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit. e –0.75x The domain is all real numbers, and the range is y > 1.

EXAMPLE 3 Solve a multi-step problem Biology The length l (in centimeters) of a tiger shark can be modeled by the function e –0.178t l = 337 – 276 where t is the shark’s age (in years). • Graph the model. • Use the graph to estimate the length of a tiger shark that is 3 years old.

EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Graph the model, as shown. STEP 2 Use the trace feature to determine that l 175 when t = 3. The length of a 3year-old tiger shark is about 175 centimeters. ANSWER

GUIDED PRACTICE Graph the function. State the domain and range. 1. y = 2e 0.5x domain: all real numbers, range: y>0 ANSWER

GUIDED PRACTICE Graph the function. State the domain and range. 1 2 2. f(x) = e -x +1 domain: all real numbers, range: y>1 ANSWER

GUIDED PRACTICE Graph the function. State the domain and range. 3. y = 1.5e 0.25(x−1) −2 domain: all real numbers, range: y > −2 ANSWER

Model continuously compounded interest EXAMPLE 4 Model continuously compounded interest Finance You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year? SOLUTION Use the formula for continuously compounded interest. A = Pert Write formula. = 4000 e0.06(1) Substitute 4000 for P, 0.06 for r, and 1 for t. 4247.35 Use a calculator. The balance at the end of 1 year is $4247.35. ANSWER

GUIDED PRACTICE 4. FINANCE: You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time? a. 2 years The balance at the end of 2 years is $2762.93. ANSWER b. 5 years The balance at the end of 5 years is $3210.06. ANSWER c. 7.5 years The balance at the end of 7.5 years is $3637.48. ANSWER

GUIDED PRACTICE 4. FINANCE: Find the amount of interest earned in parts (a) – (c) of Exercise 4. a. $262.93 ANSWER $710.06 b. $1137.48 c.