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College Algebra Chapter 4 Exponential and Logarithmic Functions
Section 4.2 Exponential Functions Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
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Concepts Graph Exponential Functions
Evaluate the Exponential Function Base e Use Exponential Functions to Compute Compound Interest Use Exponential Functions in Applications
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Concept 1 Graph Exponential Functions
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Graph Exponential Functions (1 of 2)
Previously: Now:
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Graph Exponential Functions (2 of 2)
Exponential Function: Let b be a constant real number such that b > 0 and b ≠ 1. Then for any real number x, a function of the form is called an exponential function of base b.
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Example 1 (1 of 2) Graph the function. Solution:
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Example 1 (2 of 2)
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Example 2 (1 of 2) Graph the function. Solution:
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Example 2 (2 of 2)
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Skill Practice 1 Graph the functions.
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Example 3 (1 of 4) Graph the function. Solution:
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Example 3 (2 of 4)
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Example 3 (3 of 4) Parent function: Where in, ab:
If a<0 reflect across the x-axis. Shrink vertically if 0< |a| < 1. Stretch vertically if |a| > 1. x – h: If h > 0, shift to the right. If h < 0, shift to the left.
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Example 3 (4 of 4) k: If k > 0, shift upward.
If k < 0, shift downward. Where, x + 1: 1 to left -2: reflect across x-axis 1: 1 up
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Skill Practice 2 Graph.
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Concept 2 Evaluate the Exponential Function Base e
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Evaluate the Exponential Function Base e
e is a universal constant (like the number ) and an irrational number. Approaches a constant value that we call e e ≈
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Examples 4 – 7 Evaluate. Round to 4 decimal places.
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Example 8 (1 of 2) Graph the function. Solution:
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Example 8 (2 of 2)
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Skill Practice 3
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Concept 3 Use Exponential Functions to Compute Compound Interest
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Use Exponential Functions to Compute Compound Interest
Suppose that P dollars in principle is invested (or borrowed) at an annual interest rate r for t years. Then: I = Prt: Amount of simple interest I (in $) Amount A (in $) in the account after t years under continuous compounding. Amount A (in $) in the account after t years and n compounding periods per year.
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Example 9 (1 of 2) Suppose that $15,000 is invested with 2.5% interest under the following compounding options. Determine the amount in the account at the end of 7 years for each option. Compounded annually Solution:
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Example 9 (2 of 2) Compounded quarterly Solution:
Compounded continuously Solution:
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Skill Practice 4 Suppose that $8000 is invested and pays 4.5% per year under the following compounding options. Compounded annually Compounded quarterly Compounded monthly Compounded daily Compounded continuously Determine the total amount in the account after 5 year with each option.
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Concept 4 Use Exponential Functions in Applications
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Example 10 (1 of 3) Weapon-grade plutonium is composed of approximately 93% plutonium-239 (Pu-239). The half-life of Pu-239 is 24,000 years. In a sample originally containing 0.5 kilograms, the amount left after t years is given by Evaluate the function for the given values of t and interpret the meaning in context.
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Example 10 (2 of 3) A(24,000) Solution:
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Example 10 (3 of 3) A(72,000) Solution:
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Example 11 (1 of 2) A 2010 estimate of the population of Mexico is 111 million people with a projected growth rate of 0.994% per year. At this rate, the population P(t) (in millions) can be approximated by where t is the time in years since 2010.
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Example 11 (2 of 2) Evaluate P(4) and interpret its meaning in the context of this problem. Solution: In 2014 population of Mexico will be ≈ 115 million. Evaluate P(75) and interpret its meaning in the context of this problem. Solution: In 2085 population of Mexico will be ≈ 233 million.
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Skill Practice 5 Cesium-137 is a radioactive metal with a short half-life of 30 yr. In a sample originally having 2 g of cesium-137, the amount A(t) (in grams) of cesium-137 present after t years is given by How much cesium-137 will be present after 30 year? 60 year? 90 year?
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