Lesson 3.1

Lesson 3.1 Answer: Polynomials and rational functions are examples of algebraic functions. Answer: Exponential and logarithmic functions are examples of nonalgebraic functions, also called transcendental functions. Answer: You can use the one-to-one property to solve simple exponential equations. Answer: The exponential function f (x) = ex is called the natural exponential function, and the base e is called the natural base. Answer: To find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula A = P ¿ . Answer: To find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula A = Pe rt .

Answer: 200 ¿ ¿ 200 × ¿ ¿ 1.274174347 × 10 25 1.274 × 10 25 Answer:

Answer: Answer: x y -3 0.125 -2 0.25 -1 0.5 0 1 1 2 2 4 3 8 X = -3 -5 0 5

Answer: x y -2 3073 1024 -1 769 256 0 193 64 1 49 16 2 13 4 -5 0 5

Answer: The graph of f is: 0 2 4 6 8 The transformed function is g ( x ) =− 0.3 x + 5 , it can written as: g ( x ) =− f ( x ) + 5 5 -5 0 5 -5

Answer: f ( 240 ) = 1.5 e 240 2 ¿ 1.5 e 120 Therefore, f ( 240 ) = 1.956 × 10 52 Answer: x y -2 e 2 -1 e 0 1 1 1 e 2 1 e 2

Answer: 2 x − 1 = 4 2 x − 1 + 1 = 4 + 1 2 x = 5 2 x 2 = 5 2 x = 5 2 The value of x in the equation e 2 x − 1 e 4 is 5 2 Answer: x 2 + 6 = 5 x x 2 − 5 x + 6 = 0 x 2 − 3 x − 2 x + 6 = 0 x ( x − 3 ) − 2 ( x − 3 ) = 0 ( x − 3 ) ( x − 2 ) = 0 x − 3 = 0 and x − 2 = 0 x = 3 and x = 2 The values of x in the equation e x 2 + 6 = e 5 x

Answer: n 1 2 4 12 365 Continuo us A $3526.5 0 $3536.9 5 $3542.2 7 $3545.8 6 $3547.6 1 $3547.6 7 Answer: t 10 20 30 40 50 A 21865.42 39841.4 72595.77 132278.1 2 241026.4 4

V ( t ) = 10000.298 (b) For t = 1,5 hour V ( t ) = 100 e 4.6052 t ¿ 100 e 4.6052 ( 1.5 ) V ( t ) = 100004.472 (c) For t = 2 hour V ( t ) = 100 e 4.6052 t ¿ 100 e 4.6052 ( 2 ) V ( t ) = 1,000,059.630

Answer: (a) Considering the equation P = 57.563 e 0.0052 t A function y = f ( x ) is said to be an increasing function if y increases as x increases. Here, as the variable t increases, the function P also increases. So the population of Italy as per the model given above is increasing. (b) The population of Italy in the year 2000 is got by substituting t = 0 in the equation. So, when t = 0, P = 57.563 e 0.0052 × 0 ¿ 57.563 × 1 ¿ 57.563 The population of Italy in the year 2000 is 57.563 millions. The population of Italy in the year 2012 is got by substituting t = 12 in the equation. So, when t = 12, P = 57.563 e 0.0052 × 12 ¿ 57.563 × 1.0644

(a) Q = 10 ¿ = 10 ¿ ¿ 10 ¿ = 10 ( 1 ) The initial quantity is Q = 10 grams. (b) Q = 10 ¿ = 10 ¿ ¿ 10 ¿ ¿ 10 ( 0.7846 ) Therefore, after 2000 years the quantity is Q = 7.846 grams (c) 10

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