Exponential Functions 2010 Pearson Education Inc GoldsteinSchneiderLayAsmar CALCULUS

Exponential Function • Definition Exponential Function: A function whose exponent is the independent variable

Properties of Exponential Functions © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS,

Simplifying Exponential Expressions EXAMPLE Write each function in the form 2 kx or 3

Graphs of Exponential Functions Notice that, no matter what b is (except 1), the

Solving Exponential Equations EXAMPLE Solve the following equation for x. SOLUTION This is the

The Exponential Function e x © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

The Number e Definition Example e: An irrational number, approximately equal to 2. 71828,

The Derivative of 2 x m = Ln 2 = 0. 693 © 2010

Solving Exponential Equations EXAMPLE Calculate. SOLUTION © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND

The Derivatives of bx and ex m = Ln b m = Ln e

Solving Exponential Equations EXAMPLE Find the equation of the tangent line to the curve

Solving Exponential Equations CONTINUED Simplify. Factor. Simplify the numerator. Now we evaluate the derivative

Solving Exponential Equations CONTINUED Now we know a point on the tangent line, (0,

Differentiation of Exponential Functions © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS,

Chain Rule for eg(x) © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS,

Chain Rule for eg(x) EXAMPLE Differentiate. SOLUTION This is the given function. Use the

Functions of the form ekx © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

The Natural Logarithmic Function © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS,

The Natural Logarithm of x Definition Example Natural logarithm of x: Given the graph

Properties of the Natural Logarithm © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

Properties of the Natural Logarithm 1) The point (1, 0) is on the graph

Exponential Expressions EXAMPLE Simplify. SOLUTION Using properties of the exponential function, we have ©

Solving Exponential Equations EXAMPLE Solve the equation for x. SOLUTION This is the given

Solving Logarithmic Equations EXAMPLE Solve the equation for x. SOLUTION This is the given

Other Exponential and Logarithmic Functions © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

Common Logarithms Definition Example Common logarithm: Logarithms to the base 10 © 2010 Pearson

The Derivative of ln x © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

Derivative Rules for Natural Logarithms © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Differentiate. Use the

Differentiating Logarithmic Expressions EXAMPLE The function has a relative extreme point for x >

Differentiating Logarithmic Expressions CONTINUED The derivative will equal 0 when the numerator equals 0

Differentiating Logarithmic Expressions CONTINUED Simplify. Factor and cancel. Evaluate the second derivative at x

Properties of the Natural Logarithm Function © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND

Simplifying Logarithmic Expressions EXAMPLE Write as a single logarithm. SOLUTION This is the given

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Rewrite using LIII.

Differentiating Logarithmic Expressions CONTINUED Distribute. Finish differentiating. Simplify. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar,

Logarithmic Differentiation Definition Example Logarithmic Differentiation: Given a function y = f (x), take

Logarithmic Differentiation EXAMPLE Use logarithmic differentiation to differentiate the function. SOLUTION This is the

Logarithmic Differentiation CONTINUED Use LIV. Differentiate. Solve for f ΄(x). Substitute for f (x).

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Exponential Function • Definition Exponential Function: A function whose exponent is the independent variable © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 2 of 55

Simplifying Exponential Expressions EXAMPLE Write each function in the form 2 kx or 3 kx, for a suitable constant k. SOLUTION (a) We notice that 81 is divisible by 3. And through investigation we recognize that 81 = 34. Therefore, we get (b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 4 of 55

Graphs of Exponential Functions Notice that, no matter what b is (except 1), the graph of y = bx has a y-intercept of 1. Also, if 0 < b < 1, the function is decreasing. If b > 1, then the function is increasing. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 5 of 55

Solving Exponential Equations EXAMPLE Solve the following equation for x. SOLUTION This is the given equation. Factor. Simplify. Since 5 x and 6 – 3 x are being multiplied, set each factor equal to zero. 5 x ≠ 0. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 6 of 55

The Number e Definition Example e: An irrational number, approximately equal to 2. 71828, such that the function f (x) = bx has a slope of 1, at x = 0, when b = e © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 8 of 55

Solving Exponential Equations EXAMPLE Find the equation of the tangent line to the curve at (0, 1). SOLUTION We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation. This is the given function. Differentiate. Use the quotient rule. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 12 of 55

Solving Exponential Equations CONTINUED Simplify. Factor. Simplify the numerator. Now we evaluate the derivative at x = 0. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 13 of 55

Solving Exponential Equations CONTINUED Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line. This is the point-slope form of a line. (x 1, y 1) = (0, 1) and m = -1. Simplify. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 14 of 55

Chain Rule for eg(x) EXAMPLE Differentiate. SOLUTION This is the given function. Use the chain rule. Remove parentheses. Use the chain rule for exponential functions. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 17 of 55

The Natural Logarithm of x Definition Example Natural logarithm of x: Given the graph of y = ex, the reflection of that graph about the line y = x, denoted y = ln x © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 20 of 55

Properties of the Natural Logarithm 1) The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = ex]. 2) ln x is defined only for positive values of x. 3) ln x is negative for x between 0 and 1. 4) ln x is positive for x greater than 1. 5) ln x is an increasing function and concave down. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 22 of 55

Exponential Expressions EXAMPLE Simplify. SOLUTION Using properties of the exponential function, we have © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 23 of 55

Solving Exponential Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Remove the parentheses. Combine the exponential expressions. Add. Take the logarithm of both sides. Simplify. Finish solving for x. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 24 of 55

Solving Logarithmic Equations EXAMPLE Solve the equation for x. SOLUTION This is the given equation. Divide both sides by 5. Rewrite in exponential form. Divide both sides by 2. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 25 of 55

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Differentiate. Use the power rule. Differentiate ln[g(x)]. Finish. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 30 of 55

Differentiating Logarithmic Expressions EXAMPLE The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point? SOLUTION This is the given function. Use the quotient rule to differentiate. Simplify. Set the derivative equal to 0. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 31 of 55

Differentiating Logarithmic Expressions CONTINUED The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0. Set the numerator equal to 0. Write in exponential form. To determine whether the function has a relative maximum at x = 1, let’s use the second derivative. This is the first derivative. Differentiate. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 32 of 55

Differentiating Logarithmic Expressions CONTINUED Simplify. Factor and cancel. Evaluate the second derivative at x = 1. Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point So, the relative maximum occurs at (1, 1). © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 33 of 55

Simplifying Logarithmic Expressions EXAMPLE Write as a single logarithm. SOLUTION This is the given expression. Use LIV (this must be done first). Use LIII. Use LI. Simplify. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 36 of 55

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Rewrite using LIII. Rewrite using LIV. Differentiate. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 37 of 55

Differentiating Logarithmic Expressions CONTINUED Distribute. Finish differentiating. Simplify. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 38 of 55

Logarithmic Differentiation Definition Example Logarithmic Differentiation: Given a function y = f (x), take the natural logarithm of both sides of the equation, Example will follow. use logarithmic rules to break up the right side of the equation into any number of factors, differentiate each factor, and finally solving for the desired derivative. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 39 of 55

Logarithmic Differentiation EXAMPLE Use logarithmic differentiation to differentiate the function. SOLUTION This is the given function. Take the natural logarithm of both sides of the equation. Use LIII. Use LI. © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 40 of 55

Logarithmic Differentiation CONTINUED Use LIV. Differentiate. Solve for f ΄(x). Substitute for f (x). © 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12 e– Slide 41 of 55