October 15 MATH 1113 sec. 51 Fall 2018

Transcription

1 October 15 MATH 1113 sec. 51 Fall 2018 Section 5.5: Solving Exponential and Logarithmic Equations Base-Exponent Equality For any a > 0 with a 1, and for any real numbers x and y a x = a y if and only if x = y. Logarithm Equality For and a > 0 with a 1, and for any positive numbers x and y log a x = log a y if and only if x = y. Inverse Function For any a > 0 with a 1 a log a x = x for every x > 0 log a (a x ) = x for every real x. October 12, / 42

2 Example Solve the equation 3 2x+1 = 81. Show that the same result is obtained by two approaches. (a) by using the fact that 81 = 3 4 and equating exponents. October 12, / 42

3 Example Solve the equation 3 2x+1 = 81. Show that the same result is obtained by two approaches. (b) by using the base 3 logarithm as an inverse function. October 12, / 42

4 Example Find an exact solution 1 to the equation 2 x+1 = 5 x 1 An exact solution may be a number such as 2 or ln(7) which requires a calculator to approximate as a decimal. October 12, / 42

5 Graphical Solution to 2 x+1 = 5 x Figure: Plots of y = 2 x+1 and y = 5 x together. The curves intersect at the solution x = ln 2/(ln 5 ln 2) Which curve is y = 2 x+1, red or blue? October 12, / 42

6 Question An exact solution to 3 x = 4 x 1 can be found using the natural logarithm. An exact solution is (a) x = (b) x = (c) x = (d) x = ln 4 ln 4 ln 3 ln 3 ln 4 ln 3 ln 4 ln 4 + ln 3 ln 3 ln 4 + ln 3 (e) I know how to do this, but my answer is not here October 12, / 42

7 An Observation To solve 2 x+1 = 5 x, we used the natural log. But we have choices. Use the change of base formula to show that our solution ln 2 ln 5 ln 2 = log 2 log 5 log 2 October 12, / 42

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9 Question Jack and Diane are solving 3 x = 4 x 1. They arrive at the solutions Jack s ln 4 ln 4 + ln 3 Diane s log 3 (4) log 3 (4) + 1. Which of the following statements is true? (a) Jack s answer is correct, and Diane s is incorrect. (b) Diane s answer is correct, and Jack s is incorrect. (c) Both answers are correct; they are the same number. (d) Both answers are incorrect. October 12, / 42

10 Log Equations & Verifying Answers Double checking answers is always recommended. When dealing with functions whose domains are restricted, answer verification is critical. Use properties of logarithms to solve the equation log(x 1) + log(x 2) = log 12 October 12, / 42

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12 Question Solve the equation log 6 x + log 6 (x 1) = 1. (Hint: log 6 6 = 1.) (a) x = 3 or x = 2 (b) x = 2 or x = 3 (c) x = 3 (d) x = 2 (e) x = 0 or x = 1 October 12, / 42

13 Combining Skills Find all solutions of the equation 2 e x + e x 2 = 2 2 The function f (x) = ex +e x 2 is a well known function called the hyperbolic cosine. October 12, / 42

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16 Applications When a quantity Q changes in time according to the model Q(t) = Q 0 e kt, it is said to experience exponential growth (k > 0) or exponential decay (k < 0). Here, Q 0 is the constant initial quantity Q(0), and k is a constant. Examples of such phenomena include population changes (over small time frame), continuously compounded interest (on deposit or loan amount), substances subject to radio active decay Generally, the model for exponential decay is written Q(t) = Q 0 e kt, and for growth it is written as Q(t) = Q 0 e kt so that it is always assumed that k > 0. October 12, / 42

17 Example The 44 Ti titanium isotope decays to 44 Ca, a stable calcium isotope. The mass Q is subject to exponential decay Q(t) = Q 0 e kt for t in years, and some k > 0. If the half life (amount of time for the mass to reduce by 50%) is 60 years, determine the value of k. October 12, / 42

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19 Combining Skills The function S h (x) = ex e x is one to one. It has a special name; 2 it s called the hyberbolic sine function. Find its inverse function S 1 h (x). Figure: Plot of the hyperbolic sine function. Note that it has odd symmetry and looks a little like a cubic or odd power function. October 12, / 42

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22 Sections 6.1 & 6.2: Trigonometric Functions of Acute Angles In this section, we are going to define six new functions called trigonometric functions. We begin with an acute angle θ in a right triangle with the sides whose lengths are labeled: October 12, / 42

23 Sine, Cosine, and Tangent For the acute angle θ, we define the three numbers as follows sin θ = opp hyp, cos θ = adj hyp, tan θ = opp adj, read as sine theta read as cosine theta read as tangent theta Note that these are numbers, ratios of side lengths, and have no units. It may be convenient to enclose the argument of a trig function in parentheses. That is, sin θ = sin(θ). October 12, / 42

24 Cosecant, Secant, and Cotangent The remaining three trigonometric functions are the reciprocals of the first three csc θ = hyp opp = 1 sin θ, sec θ = hyp adj = 1 cos θ, cot θ = adj opp = 1 tan θ, read as cosecant theta read as secant theta read as cotangent theta October 12, / 42

25 A Word on Notation The trigonometric ratios define functions: input angle number output ratio number. From the definitions, we see that csc θ = 1 sin θ. Functions have arguments. It is NOT acceptable to write the above relationship as csc = 1 sin. October 12, / 42

26 Example Determine the six trigonometric values of the acute angle θ. October 12, / 42

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28 Example Determine the six trigonometric values of the acute angle θ. October 12, / 42

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30 Question For the angle θ shown, which statement is correct? October 12, / 42

31 Some Key Trigonometric Values Use the triangles to determine the six trigonometric values of the angles 30, 45, and 60. Figure: An isosceles right triangle of leg length 1 (left), and half of an equilateral triangle of side length 2 (right). October 12, / 42

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33 Commit To Memory It is to our advantage to remember the following: sin 30 = 1 2, sin 45 = 1 2, sin 60 = 3 2 cos 30 = 3 2, cos 45 = 1 2, cos 60 = 1 2 tan 30 = 1 3, tan 45 = 1, tan 60 = 3 We ll use these to find some other trigonometric values. Still others will require a calculator. October 12, / 42

34 Calculator Figure: Any scientific calculator will have built in functions for sine, cosine and tangent. (TI-84 shown) October 12, / 42

35 Using a Calculator Evaluate the following using a calculator. Round answers to three decimal places. sin 16 = sec 78.3 = tan(65.4 ) = October 12, / 42

36 Application Example Before cutting down a dead tree, you wish to determine its height. From a horizontal distance of 40 ft, you measure the angle of elevation from the ground to the top of the tree to be 61. Determine the tree height to the nearest 100 th of a foot. October 12, / 42

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38 Application Example The ramp of truck for moving touches the ground 14 feet from the end of the truck. If the ramp makes an angle of 28.5 with the ground, what is the length of the ramp? October 12, / 42

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40 Complementary Angles and Cofunction Identities The two acute angles in a right triangle must sum to 90. Two acute angles whose measures sum to 90 are called complementary angles. Given an acute angle θ its complement is the angle 90 θ. Example Find the complementary angle of 27. October 12, / 42

41 Cofunction Identities Figure: Note that for complementary angles θ and φ, the role of the legs (opposite versus adjacent) are interchanged. October 12, / 42

42 Cofunction Identities For any acute angle θ sin θ = cos(90 θ) cos θ = sin(90 θ) tan θ = cot(90 θ) cot θ = tan(90 θ) sec θ = csc(90 θ) csc θ = sec(90 θ) These equations define what are called cofunction identities. October 12, / 42