2.1 the Derivative and the Tangent Line Problem

Home , Difference quotient, Graph of a function, Secant line, Vertical tangent

96 Chapter 2 Differentiation

2.1 The Derivative and the Tangent Line Problem

Find the slope of the tangent line to a curve at a point. Use the limit definition to find the derivative of a function. Understand the relationship between differentiability and continuity.

The Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem (Section 1.1 and this section) 2. The velocity and acceleration problem (Sections 2.2 and 2.3) 3. The minimum and maximum problem (Section 3.1) 4. The area problem (Sections 1.1 and 4.2) Each problem involves the notion of a limit, and calculus can be introduced with any of the four problems. A brief introduction to the tangent line problem is given in Section 1.1. Although partial solutions to this problem were given by Pierre de Fermat (1601–1665), René Descartes (1596–1650), Christian Huygens (1629–1695), and Isaac Barrow (1630–1677), credit for the first general solution is usually given to Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716). Newton’s work on this problem ISAAC NEWTON (1642–1727) stemmed from his interest in optics and light refraction. In addition to his work in calculus, What does it mean to say that a line is Newton made revolutionary y contributions to physics, including tangent to a curve at a point? For a circle, the the Law of Universal Gravitation tangent line at a point P is the line that is and his three laws of motion. perpendicular to the radial line at point P, as See LarsonCalculus.com to read shown in Figure 2.1. more of this biography. For a general curve, however, the problem P is more difficult. For instance, how would you define the tangent lines shown in Figure 2.2? You might say that a line is tangent to a curve at a point P when it touches, but does not cross, the curve at point P. This definition would work x for the first curve shown in Figure 2.2, but not for the second. Or you might say that a line is tangent to a curve when the line touches or intersects the curve at exactly one point. This Tangent line to a circle definition would work for a circle, but not for Figure 2.1 more general curves, as the third curve in Figure 2.2 shows.

Exploration y y y Use a graphing utility to graph f ͑x͒ ϭ 2x3 Ϫ 4x2 ϩ 3x Ϫ 5. y = f(x) P On the same screen, graph P ϭ Ϫ ϭ Ϫ y = f(x) y = f(x) y x 5, y 2x 5, and P y ϭ 3x Ϫ 5. Which of these lines, if any, appears to be tangent to the graph of f at x x x the point ͑0, Ϫ5͒? Explain your reasoning. Tangent line to a curve at a point Figure 2.2 Mary Evans Picture Library/Alamy

y Essentially, the problem of finding the tangent line at a point P boils down to the problem of finding the slope of the tangent line at point P. You can approximate this slope using a secant line* through the point of tangency and a second point on the ͑ ͑ ͒͒ (c + Δx, f(c + Δx)) curve, as shown in Figure 2.3. If c, f c is the point of tangency and ͑c ϩ⌬x, f ͑c ϩ⌬x͒͒ f(c + Δx) − f(c) = Δy is a second point on the graph of f, then the slope of the secant line through the two (c, f(c)) points is given by substitution into the slope formula Δ x y Ϫ y x ϭ 2 1 m Ϫ x2 x1 ͑ ͑ ͒͒ The secant line through c, f c and f͑c ϩ⌬x͒ Ϫ f ͑c͒ Change in y ͑ ϩ⌬ ͑ ϩ⌬ ͒͒ m ϭ c x, f c x sec ͑c ϩ⌬x͒ Ϫ c Change in x Figure 2.3 f͑c ϩ⌬x͒ Ϫ f ͑c͒ m ϭ . Slope of secant line sec ⌬x

The right-hand side of this equation is a difference quotient. The denominator ⌬x is the change in x, and the numerator ⌬y ϭ f ͑c ϩ⌬x͒ Ϫ f ͑c͒ is the change in y. The beauty of this procedure is that you can obtain more and more accurate approximations of the slope of the tangent line by choosing points closer and closer to the point of tangency, as shown in Figure 2.4.

THE TANGENT LINE PROBLEM Δx → 0 (c, f(c)) Δ In 1637, mathematician René y Descartes stated this about the Δy (c, f(c)) tangent line problem: Δx c, f c Δx ( ( )) Δy “And I dare say that this is not Δy (c, f(c)) only the most useful and general Δx Δx problem in geometry that I know, (c, f(c)) (c, f(c)) Δy but even that I ever desire to know.” Δ y Δx Δx (c, f(c)) (c, f(c)) Δx → 0 Tangent line Tangent line Tangent line approximations Figure 2.4

Definition of Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit ⌬y f ͑c ϩ⌬x͒ Ϫ f ͑c͒ lim ϭ lim ϭ m ⌬x→0 ⌬x ⌬x→0 ⌬x exists, then the line passing through ͑c, f ͑c͒͒ with slope m is the tangent line to the graph of f at the point ͑c, f ͑c͒͒.

The slope of the tangent line to the graph of f at the point ͑c, f ͑c͒͒ is also called the .c ؍ slope of the graph of f at x

* This use of the word secant comes from the Latin secare, meaning to cut, and is not a reference to the trigonometric function of the same name.

The Slope of the Graph of a Linear Function

To find the slope of the graph of f ͑x͒ ϭ 2x Ϫ 3 when c ϭ 2, you can apply the y f(x) = 2x − 3 definition of the slope of a tangent line, as shown. ͑ ϩ⌬ ͒ Ϫ ͑ ͒ ͓ ͑ ϩ⌬ ͒ Ϫ ͔ Ϫ ͓ ͑ ͒ Ϫ ͔ Δx = 1 f 2 x f 2 ϭ 2 2 x 3 2 2 3 3 lim lim ⌬x→0 ⌬x ⌬x→0 ⌬x ϩ ⌬ Ϫ Ϫ ϩ ϭ 4 2 x 3 4 3 lim 2 Δy = 2 ⌬x→0 ⌬x 2⌬x m = 2 ϭ lim ⌬x→0 ⌬x 1 (2, 1) ϭ lim 2 ⌬x→0 x ϭ 2 123 The slope of f at ͑c, f ͑c͒͒ ϭ ͑2, 1͒ is m ϭ 2, as shown in Figure 2.5. Notice that the limit The slope of f at ͑2, 1͒ is m ϭ 2. definition of the slope of f agrees with the definition of the slope of a line as discussed Figure 2.5 in Section P.2.

The graph of a linear function has the same slope at any point. This is not true of nonlinear functions, as shown in the next example.

Tangent Lines to the Graph of a Nonlinear Function

y Find the slopes of the tangent lines to the graph of f ͑x͒ ϭ x2 ϩ 1 at the points ͑0, 1͒ and ͑Ϫ1, 2͒, as shown in Figure 2.6. 4 Solution Let ͑c, f ͑c͒͒ represent an arbitrary point on the graph of f. Then the slope 3 of the tangent line at ͑c, f ͑c͒͒ can be found as shown below. [Note in the limit process 2 f(x) = x + 1 that c is held constant (as ⌬x approaches 0).] Tangent 2 line at Tangent line f ͑c ϩ⌬x͒ Ϫ f ͑c͒ ͓͑c ϩ⌬x͒ 2 ϩ 1͔ Ϫ ͑c2 ϩ 1͒ (−1, 2) at (0, 1) lim ϭ lim ⌬x→0 ⌬x ⌬x→0 ⌬x c2 ϩ 2c͑⌬x͒ ϩ ͑⌬x͒2 ϩ 1 Ϫ c2 Ϫ 1 x ϭ lim −2 −1 1 2 ⌬x→0 ⌬x The slope of f at any point ͑c, f͑c͒͒ is 2c͑⌬x͒ ϩ ͑⌬x͒2 ϭ lim m ϭ 2c. ⌬x→0 ⌬x Figure 2.6 ϭ lim ͑2c ϩ⌬x͒ ⌬x→0 ϭ 2c So, the slope at any point ͑c, f ͑c͒͒ on the graph of f is m ϭ 2c. At the point ͑0, 1͒, the y Vertical slope is m ϭ 2͑0͒ ϭ 0, and at ͑Ϫ1, 2͒, the slope is m ϭ 2͑Ϫ1͒ ϭϪ2. tangent line The definition of a tangent line to a curve does not cover the possibility of a vertical tangent line. For vertical tangent lines, you can use the following definition. If f is continuous at c and (c, f(c)) f ͑c ϩ⌬x͒ Ϫ f ͑c͒ f ͑c ϩ⌬x͒ Ϫ f ͑c͒ lim ϭ ϱ or lim ϭϪϱ ⌬x→0 ⌬x ⌬x→0 ⌬x

x then the vertical line x ϭ c passing through ͑c, f ͑c͒͒ is a vertical tangent line to the c graph of f. For example, the function shown in Figure 2.7 has a vertical tangent line at The graph of f has a vertical tangent ͑c, f ͑c͒͒. When the domain of f is the closed interval ͓a, b͔, you can extend the line at ͑c, f͑c͒͒. definition of a vertical tangent line to include the endpoints by considering continuity Figure 2.7 and limits from the right ͑for x ϭ a͒ and from the left ͑for x ϭ b͒.

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.1 The Derivative and the Tangent Line Problem 99 The Derivative of a Function You have now arrived at a crucial point in the study of calculus. The limit used to define the slope of a tangent line is also used to define one of the two fundamental operations of calculus—differentiation.

Definition of the Derivative of a Function The derivative of f at x is f ͑x ϩ⌬x͒ Ϫ f ͑x͒ fЈ͑x͒ ϭ lim ⌬x→0 ⌬x REMARK The notation fЈ͑x͒ Ј is read as “f prime of x. ” provided the limit exists. For all x for which this limit exists,f is a function of x.

Be sure you see that the derivative of a function of x is also a function of x. This “new” function gives the slope of the tangent line to the graph of f at the point ͑x, f ͑x͒͒, provided that the graph has a tangent line at this point. The derivative can also be used to determine the instantaneous rate of change (or simply the rate of change) of one variable with respect to another. FOR FURTHER INFORMATION The process of finding the derivative of a function is called differentiation. A For more information on the function is differentiable at x when its derivative exists at x and is differentiable on an crediting of mathematical discoveries open interval ͧa, bͨ when it is differentiable at every point in the interval. to the first “discoverers,” see the In addition to fЈ͑x͒, other notations are used to denote the derivative of y ϭ f ͑x͒. article “Mathematical Firsts— The most common are Who Done It?” by Richard H. Williams and Roy D. Mazzagatti in Ј͑ ͒ dy Ј d ͓ ͑ ͔͒ ͓ ͔ Mathematics Teacher. To view this f x , , y , f x , Dx y . Notation for derivatives article, go to MathArticles.com. dx dx

The notation dy͞dx is read as “the derivative of y with respect to x” or simply “dy, dx.” Using limit notation, you can write

⌬ ͑ ϩ⌬ ͒ Ϫ ͑ ͒ dy ϭ y ϭ f x x f x ϭ Ј͑ ͒ lim lim f x . dx ⌬x→0 ⌬x ⌬x→0 ⌬x

Finding the Derivative by the Limit Process

See LarsonCalculus.com for an interactive version of this type of example. To find the derivative of f ͑x͒ ϭ x3 ϩ 2x, use the definition of the derivative as shown. f ͑x ϩ⌬x͒ Ϫ f ͑x͒ fЈ͑x͒ ϭ lim Definition of derivative ⌬x→0 ⌬x ͑x ϩ⌬x͒3 ϩ 2͑x ϩ⌬x͒ Ϫ ͑x3 ϩ 2x͒ ϭ lim ⌬ → ⌬ REMARK When using the x 0 x 3 ϩ 2⌬ ϩ ͑⌬ ͒2 ϩ ͑⌬ ͒3 ϩ ϩ ⌬ Ϫ 3 Ϫ definition to find a derivative of ϭ x 3x x 3x x x 2x 2 x x 2x lim a function, the key is to rewrite ⌬x→0 ⌬x the difference quotient so that 2⌬ ϩ ͑⌬ ͒2 ϩ ͑⌬ ͒3 ϩ ⌬ ϭ 3x x 3x x x 2 x ⌬x lim does not occur as a factor ⌬x→0 ⌬x of the denominator. ⌬ ͓ 2 ϩ ⌬ ϩ ͑⌬ ͒2 ϩ ͔ ϭ x 3x 3x x x 2 lim ⌬x→0 ⌬x ϭ ͓ 2 ϩ ⌬ ϩ ͑⌬ ͒2 ϩ ͔ lim 3x 3x x x 2 ⌬x→0 ϭ 3x2 ϩ 2

Using the Derivative to Find the Slope at a Point REMARK Remember that Find fЈ͑x͒ for f ͑x͒ ϭ Ίx. Then find the slopes of the graph of f at the points ͑1, 1͒ and the derivative of a function f is ͑4, 2͒. Discuss the behavior of f at ͑0, 0͒. itself a function, which can be used to find the slope of the Solution Use the procedure for rationalizing numerators, as discussed in Section 1.3. tangent line at the point f ͑x ϩ⌬x͒ Ϫ f ͑x͒ ͑x, f ͑x͒͒ on the graph of f. fЈ͑x͒ ϭ lim Definition of derivative ⌬x→0 ⌬x Ί ϩ⌬ Ϫ Ί ϭ x x x lim ⌬x→0 ⌬x Ί ϩ⌬ Ϫ Ί Ί ϩ⌬ ϩ Ί ϭ x x x x x x lim ΂ ΃΂ ΃ y ⌬x→0 ⌬x Ίx ϩ⌬x ϩ Ίx ͑ ϩ⌬ ͒ Ϫ ϭ x x x 3 lim ⌬x→0 ⌬x͑Ίx ϩ⌬x ϩ Ίx ͒ (4, 2) ⌬x ϭ lim 2 ⌬ → Ί Ί (1, 1) 1 x 0 ⌬x͑ x ϩ⌬x ϩ x ͒ m = 4 ϭ 1 lim 1 f(x) = x ⌬x→0 Ί ϩ⌬ ϩ Ί m = x x x 2 1 x ϭ , x > 0 (0, 0) 1 2 3 4 2Ίx ͑ ͒ Ј͑ ͒ ϭ 1 ͑ ͒ Ј͑ ͒ ϭ 1 The slope of f at ͑x, f͑x͒͒, x > 0, is At the point 1, 1 , the slope is f 1 2. At the point 4, 2 , the slope is f 4 4. See m ϭ 1͑͞2Ίx͒. Figure 2.8. At the point ͑0, 0͒, the slope is undefined. Moreover, the graph of f has a Figure 2.8 vertical tangent line at ͑0, 0͒.

Finding the Derivative of a Function

REMARK In many See LarsonCalculus.com for an interactive version of this type of example. applications, it is convenient ϭ ͞ to use a variable other than x Find the derivative with respect to t for the function y 2 t. as the independent variable, Solution Considering y ϭ f ͑t͒, you obtain as shown in Example 5. ͑ ϩ⌬͒ Ϫ ͑ ͒ dy ϭ f t t f t lim Definition of derivative dt ⌬t→0 ⌬t 2 Ϫ 2 t ϩ⌬t t ϭ ͑ ϩ⌬͒ ϭ 2 ͑ ͒ ϭ 2 lim f t t and f t ⌬t→0 ⌬t t ϩ⌬t t 2t Ϫ 2͑t ϩ⌬t͒ ͑ ϩ⌬͒ ϭ t t t lim Combine fractions in numerator. ⌬t→0 ⌬t Ϫ2⌬t ϭ ⌬ 2 lim Divide out common factor of t. y = ⌬t→0 ⌬t͑t͒͑t ϩ⌬t͒ 4 t Ϫ ϭ 2 lim Simplify. ⌬t→0 t͑t ϩ⌬t͒

(1, 2) 2 ϭϪ . Evaluate limit as ⌬t → 0. t2

TECHNOLOGY A graphing utility can be used to reinforce the result given 0 6 ͞ ϭϪ ͞ 2 0 y = −2t + 4 in Example 5. For instance, using the formula dy dt 2 t , you know that the slope of the graph of y ϭ 2͞t at the point ͑1, 2͒ is m ϭϪ2. Using the point-slope At the point ͑1, 2͒, the line form, you can find that the equation of the tangent line to the graph at ͑1, 2͒ is y ϭϪ2t ϩ 4 is tangent to the graph y Ϫ 2 ϭϪ2͑t Ϫ 1͒ or y ϭϪ2t ϩ 4 of y ϭ 2͞t. Figure 2.9 as shown in Figure 2.9.

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.1 The Derivative and the Tangent Line Problem 101 Differentiability and Continuity The alternative limit form of the derivative shown below is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is

f ͑x͒ Ϫ f ͑c͒ fЈ͑c͒ ϭ lim Alternative form of derivative x→c x Ϫ c REMARK A proof of the equivalence of the alternative form of the derivative is provided this limit exists (see Figure 2.10).

given in Appendix A. y See LarsonCalculus.com for Bruce Edwards’s video of this proof. (x, f(x)) (c, f(c))

x − c f(x) − f(c)

x cx As x approaches c, the secant line approaches the tangent line. Figure 2.10

Note that the existence of the limit in this alternative form requires that the one-sided limits f ͑x͒ Ϫ f ͑c͒ limϪ x→c x Ϫ c and f ͑x͒ Ϫ f ͑c͒ lim x→cϩ x Ϫ c exist and are equal. These one-sided limits are called the derivatives from the left and from the right, respectively. It follows that f is differentiable on the closed interval [a, b] when it is differentiable on ͑a, b͒ and when the derivative from the right at a and the derivative from the left at b both exist. y When a function is not continuous at x ϭ c, it is also not differentiable at x ϭ c. For instance, the greatest integer function 2 f ͑x͒ ϭ ͠x͡ 1 is not continuous at x ϭ 0, and so it is not differentiable at x ϭ 0 (see Figure 2.11). You x −2 −1 123 can verify this by observing that f(x) = [[x ]] f ͑x͒ Ϫ f ͑0͒ ͠x͡ Ϫ 0 ϭ ϭ ϱ Derivative from the left limϪ limϪ −2 x→0 x Ϫ 0 x→0 x and The greatest integer function is not differentiable at x ϭ 0 because it is f ͑x͒ Ϫ f ͑0͒ ͠x͡ Ϫ 0 ϭ ϭ Derivative from the right ϭ limϩ limϩ 0. not continuous at x 0. x→0 x Ϫ 0 x→0 x Figure 2.11 Although it is true that differentiability implies continuity (as shown in Theorem 2.1 on the next page), the converse is not true. That is, it is possible for a function to be continuous at x ϭ c and not differentiable at x ϭ c. Examples 6 and 7 illustrate this possibility.

A Graph with a Sharp Turn

See LarsonCalculus.com for an interactive version of this type of example. y The function f ͑x͒ ϭ Խx Ϫ 2Խ, shown in Figure 2.12, is continuous at x ϭ 2. The one-sided limits, however, 3 f(x) = ⏐x − 2⏐ f ͑x͒ Ϫ f ͑2͒ Խx Ϫ 2Խ Ϫ 0 lim ϭ lim ϭϪ1 Derivative from the left 2 m = −1 x→2Ϫ x Ϫ 2 x→2Ϫ x Ϫ 2

1 and m = 1 f ͑x͒ Ϫ f ͑2͒ Խx Ϫ 2Խ Ϫ 0 x lim ϭ lim ϭ 1 Derivative from the right 1 2 3 4 x→2ϩ x Ϫ 2 x→2ϩ x Ϫ 2 ϭ f is not differentiable at x 2 because are not equal. So,f is not differentiable at x ϭ 2 and the graph of f does not have a the derivatives from the left and from tangent line at the point ͑2, 0͒. the right are not equal. Figure 2.12 A Graph with a Vertical Tangent Line

y The function f ͑x͒ ϭ x1͞3 is continuous at x ϭ 0, as shown in Figure 2.13. However, f(x) = x1/3 because the limit 1 f ͑x͒ Ϫ f ͑0͒ x1͞3 Ϫ 0 1 lim ϭ lim ϭ lim ͞ ϭ ϱ x→0 x Ϫ 0 x→0 x x→0 x2 3

x is infinite, you can conclude that the tangent line is vertical at x ϭ 0. So,f is not − − 2 1 1 2 differentiable at x ϭ 0.

− 1 From Examples 6 and 7, you can see that a function is not differentiable at a point at which its graph has a sharp turn or a vertical tangent line. f is not differentiable at x ϭ 0 because f has a vertical tangent line at x ϭ 0. Figure 2.13 THEOREM 2.1 Differentiability Implies Continuity If f is differentiable at x ϭ c, then f is continuous at x ϭ c.

Proof You can prove that f is continuous at x ϭ c by showing that f ͑x͒ approaches f ͑c͒ as x → c. To do this, use the differentiability of f at x ϭ c and consider the following limit. TECHNOLOGY Some f ͑x͒ Ϫ f ͑c͒ lim ͓ f ͑x͒ Ϫ f ͑c͔͒ ϭ lim ΄͑x Ϫ c͒΂ ΃΅ graphing utilities, such as x→c x→c x Ϫ c Maple, Mathematica, and the f ͑x͒ Ϫ f ͑c͒ TI-nspire, perform symbolic ϭ ΄lim ͑x Ϫ c͒΅΄lim ΅ x→c x→c x Ϫ c differentiation. Others perform ϭ ͑ ͓͒ Ј͑ ͔͒ numerical differentiation by 0 f c finding values of derivatives ϭ 0 using the formula Because the difference f ͑x͒ Ϫ f ͑c͒ approaches zero as x → c, you can conclude that ͑ ͒ ϭ ͑ ͒ ϭ f ͑x ϩ⌬x͒ Ϫ f ͑x Ϫ⌬x͒ lim f x f c . So,f is continuous at x c. Ј͑ ͒ Ϸ x→c f x ⌬ 2 x See LarsonCalculus.com for Bruce Edwards’s video of this proof. where ⌬x is a small number such as 0.001. Can you see any The relationship between continuity and differentiability is summarized below. problems with this definition? ϭ ϭ For instance, using this 1. If a function is differentiable at x c, then it is continuous at x c. So, differentia- definition, what is the value bility implies continuity. of the derivative of f ͑x͒ ϭ ԽxԽ 2. It is possible for a function to be continuous at x ϭ c and not be differentiable at when x ϭ 0? x ϭ c. So, continuity does not imply differentiability (see Example 6).

2.1 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Estimating Slope In Exercises 1 and 2, estimate the slope 1 1 21.f ͑x͒ ϭ 22. f ͑x͒ ϭ ͧ ͨ ͧ ͨ Ϫ 2 of the graph at the points x1, y1 and x2, y2 . x 1 x y y 4 1. 2. 23.f ͑x͒ ϭ Ίx ϩ 4 24. f ͑x͒ ϭ Ίx

(x1, y1) Finding an Equation of a Tangent Line In Exercises (x , y ) 2 2 25–32, (a) find an equation of the tangent line to the graph of f (x , y ) (x1, y1) 2 2 x x at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.

25.f ͑x͒ ϭ x2 ϩ 3, ͑Ϫ1, 4͒ 26. f ͑x͒ ϭ x2 ϩ 2x Ϫ 1, ͑1, 2͒ 27.f ͑x͒ ϭ x3, ͑2, 8͒ 28. f ͑x͒ ϭ x3 ϩ 1, ͑Ϫ1, 0͒ Slopes of Secant Lines In Exercises 3 and 4, use the graph shown in the figure. To print an enlarged copy of the graph, go 29.f ͑x͒ ϭ Ίx, ͑1, 1͒ 30. f ͑x͒ ϭ Ίx Ϫ 1, ͑5, 2͒ to MathGraphs.com. 4 6 31.f ͑x͒ ϭ x ϩ , ͑Ϫ4, Ϫ5͒ 32. f͑x͒ ϭ , ͑0, 3͒ y x x ϩ 2 f 6 (4, 5) Finding an Equation of a Tangent Line In Exercises 5 33–38, find an equation of the line that is tangent to the graph 4 3 of f and parallel to the given line. 2 (1, 2) Function Line 1 x 33. f ͑x͒ ϭ x2 2x Ϫ y ϩ 1 ϭ 0 1 2 3 4 5 6 34. f ͑x͒ ϭ 2x2 4x ϩ y ϩ 3 ϭ 0 3. Identify or sketch each of the quantities on the figure. 35. f ͑x͒ ϭ x3 3x Ϫ y ϩ 1 ϭ 0 f ͑ ͒ f ͑ ͒ f ͑ ͒ Ϫ f ͑ ͒ (a)1 and 4 (b) 4 1 36. f ͑x͒ ϭ x3 ϩ 2 3x Ϫ y Ϫ 4 ϭ 0 f ͑4͒ Ϫ f ͑1͒ (c) y ϭ ͑x Ϫ 1͒ ϩ f ͑1͒ 1 4 Ϫ 1 37. f ͑x͒ ϭ x ϩ 2y Ϫ 6 ϭ 0 Ίx ͑ ͒ 4. Insert the proper inequality symbol < or > between the 1 given quantities. 38. f ͑x͒ ϭ x ϩ 2y ϩ 7 ϭ 0 Ίx Ϫ 1 f ͑4͒ Ϫ f ͑1͒ f ͑4͒ Ϫ f ͑3͒ (a) ᭿ 4 Ϫ 1 4 Ϫ 3 WRITING ABOUT CONCEPTS f ͑4͒ Ϫ f ͑1͒ (b) ᭿ fЈ͑1͒ 4 Ϫ 1 Sketching a Derivative In Exercises 39–44, sketch the graph of fЈ. Explain how you found your answer. Finding the Slope of a Tangent Line In Exercises 5–10, 39.y 40. y find the slope of the tangent line to the graph of the function at 3 the given point. f x 2 −4 −2 24 ͑ ͒ ϭ Ϫ ͑Ϫ ͒ ͑ ͒ ϭ 3 ϩ ͑Ϫ Ϫ ͒ 5.f x 3 5x, 1, 8 6. g x 2 x 1, 2, 2 1 −2 x 2 2 7.g͑x͒ ϭ x Ϫ 9, ͑2, Ϫ5͒ 8. f͑x͒ ϭ 5 Ϫ x , ͑3, Ϫ4͒ −3 −2 123 f 9.f ͑t͒ ϭ 3t Ϫ t 2, ͑0, 0͒ 10. h͑t͒ ϭ t2 ϩ 4t, ͑1, 5͒ −2 −6 −3 Finding the Derivative by the Limit Process In Exercises 11–24, find the derivative of the function by the limit 41.y 42. y process. 7 7 6 6 11.f ͑x͒ ϭ 7 12. g͑x͒ ϭϪ3 5 f ͑ ͒ ϭϪ ͑ ͒ ϭ Ϫ 4 4 f 13.f x 10x 14. f x 7x 3 3 3 15.h͑s͒ ϭ 3 ϩ 2s 16. f͑x͒ ϭ 5 Ϫ 2x 2 2 3 3 1 1 17.f ͑x͒ ϭ x2 ϩ x Ϫ 3 18. f͑x͒ ϭ x2 Ϫ 5 x x −14567123 12345678 19.f ͑x͒ ϭ x3 Ϫ 12x 20. f ͑x͒ ϭ x3 ϩ x2

WRITING ABOUT CONCEPTS (continued) 58. HOW DO YOU SEE IT? The figure shows the 43.y 44. y graph of gЈ.

6 4 y f 3 4 f 6 g′ 4 2 1 2 x x − − − x 31232 1 − − −848−4 6 4 4 6 − −2 2 −4 −6 45. Sketching a Graph Sketch a graph of a function whose derivative is always negative. Explain how you (a)gЈ͑0͒ ϭ ᭿ (b) gЈ͑3͒ ϭ ᭿ found the answer. (c) What can you conclude about the graph of g knowing Ј͑ ͒ ϭϪ8 46. Sketching a Graph Sketch a graph of a function that g 1 3? whose derivative is always positive. Explain how you (d) What can you conclude about the graph of g knowing found the answer. Ј͑Ϫ ͒ ϭ 7 that g 4 3? (e) Is g͑6͒ Ϫ g͑4͒ positive or negative? Explain. 47. Using a Tangent Line The tangent line to the graph of (f) Is it possible to find g͑2͒ from the graph? Explain. y ϭ g͑x͒ at the point ͑4, 5͒ passes through the point ͑7, 0͒. Find g͑4͒ and gЈ͑4͒. ͑ ͒ ϭ 1 2 48. Using a Tangent Line The tangent line to the graph of 59. Graphical Reasoning Consider the function f x 2 x . y ϭ h͑x͒ at the point ͑Ϫ1, 4͒ passes through the point ͑3, 6͒. (a) Use a graphing utility to graph the function and estimate Find h͑Ϫ1͒ and hЈ͑Ϫ1͒. Ј͑ ͒ Ј ͑1͒ Ј͑ ͒ Ј͑ ͒ the values off 0 , f 2 , f 1 , and f 2 . Working Backwards In Exercises 49–52, the limit repre- (b) Use your results from part (a) to determine the values of fЈ͑Ϫ1͒, fЈ͑Ϫ1͒, and fЈ͑Ϫ2͒. sents fЈͧcͨ for a function f and a number c. Find f and c. 2 (c) Sketch a possible graph of fЈ. ͓5 Ϫ 3͑1 ϩ⌬x͔͒ Ϫ 2 ͑Ϫ2 ϩ⌬x͒3 ϩ 8 49.lim 50. lim (d) Use the definition of derivative to find fЈ͑x͒. ⌬x→0 ⌬x ⌬x→0 ⌬x ͑ ͒ ϭ 1 3 Ϫx2 ϩ 36 2Ίx Ϫ 6 60. Graphical Reasoning Consider the function f x 3 x . 51.lim 52. lim x→6 x Ϫ 6 x→9 x Ϫ 9 (a) Use a graphing utility to graph the function and estimate Ј͑ ͒ Ј ͑1͒ Ј͑ ͒ Ј͑ ͒ Ј͑ ͒ the values off 0 , f 2 , f 1 , f 2 , and f 3 . Writing a Function Using Derivatives In Exercises 53 (b) Use your results from part (a) to determine the values of and 54, identify a function f that has the given characteristics. fЈ͑Ϫ1͒, fЈ͑Ϫ1͒, fЈ͑Ϫ2͒, and fЈ͑Ϫ3͒. Then sketch the function. 2 (c) Sketch a possible graph of fЈ. 53. f ͑0͒ ϭ 2; fЈ ͑x͒ ϭϪ3 for Ϫϱ < x < ϱ (d) Use the definition of derivative to find fЈ͑x͒. 54. f ͑0͒ ϭ 4; fЈ ͑0͒ ϭ 0; fЈ ͑x͒ < 0 for x < 0; fЈ ͑x͒ > 0 for x > 0 Graphical Reasoning In Exercises 61 and 62, use a Finding an Equation of a Tangent Line In Exercises 55 graphing utility to graph the functions f and g in the same and 56, find equations of the two tangent lines to the graph of viewing window, where f that pass through the indicated point. fͧx 1 0.01ͨ ؊ fͧxͨ . ؍ ϭ Ϫ 2 ͑ ͒ ϭ 2 gͧxͨ ͒ ͑ 55.f x 4x x 56. f x x 0.01 y y Label the graphs and describe the relationship between them. 5 (2, 5) 10 8 2 4 61.f ͑x͒ ϭ 2x Ϫ x 62. f ͑x͒ ϭ 3Ίx 6 3 4 2 Approximating a Derivative In Exercises 63 and 64, ͧ ͨ ͧ ͨ Јͧ ͨ 1 x evaluate f 2 and f 2.1 and use the results to approximate f 2 . −6 −4 −2 2 4 6 x 1 3 − (1, −3) 63.f ͑x͒ ϭ x͑4 Ϫ x͒ 64. f ͑x͒ ϭ x 1235 4 4

57. Graphical Reasoning Use a graphing utility to graph Using the Alternative Form of the Derivative In each function and its tangent lines at x ϭϪ1, x ϭ 0, and Exercises 65–74, use the alternative form of the derivative to ؍ x ϭ 1. Based on the results, determine whether the slopes of find the derivative at x c (if it exists). tangent lines to the graph of a function at different values of x 65.f ͑x͒ ϭ x2 Ϫ 5, c ϭ 3 66. g͑x͒ ϭ x2 Ϫ x, c ϭ 1 are always distinct. 67. f ͑x͒ ϭ x3 ϩ 2x2 ϩ 1, c ϭϪ2 (a)f ͑x͒ ϭ x2 (b) g͑x͒ ϭ x3

68. f ͑x͒ ϭ x3 ϩ 6x, c ϭ 2 Determining Differentiability In Exercises 89 and 90, ؍ 69.g͑x͒ ϭ ΊԽxԽ, c ϭ 0 70. f ͑x͒ ϭ 3͞x, c ϭ 4 determine whether the function is differentiable at x 2. 1 ͑ ͒ ϭ ͑ Ϫ ͒2͞3 ϭ x2 ϩ 1, x Յ 2 x ϩ 1, x < 2 71. f x x 6 , c 6 89.f ͑x͒ ϭ Ά 90. f ͑x͒ ϭ Ά2 ͞ 4x Ϫ 3, x 2 Ί2x, x Ն 2 72. g͑x͒ ϭ ͑x ϩ 3͒1 3, c ϭϪ3 > 73.h͑x͒ ϭ Խx ϩ 7Խ, c ϭϪ7 74. f ͑x͒ ϭ Խx Ϫ 6Խ, c ϭ 6 91. Graphical Reasoning A line with slope m passes through the point ͑0, 4͒ and has the equation y ϭ mx ϩ 4. Determining Differentiability In Exercises 75–80, (a) Write the distance d between the line and the point ͑3, 1͒ describe the x -values at which f is differentiable. as a function of m. 2 75.f ͑x͒ ϭ 76. f ͑x͒ ϭ Խx2 Ϫ 9Խ (b) Use a graphing utility to graph the function d in part (a). x Ϫ 3 Based on the graph, is the function differentiable at every y y value of m? If not, where is it not differentiable? 4 12 ͑ ͒ ϭ 2 10 92. Conjecture Consider the functions f x x and 2 g͑x͒ ϭ x3. 6 x 4 (a) Graph f and fЈ on the same set of axes. 246 2 Ј −2 x (b) Graph g and g on the same set of axes. − − 4 2 2 4 (c) Identify a pattern between f and g and their respective −4 −4 derivatives. Use the pattern to make a conjecture about Ј͑ ͒ ͑ ͒ ϭ n Ն x2 h x if h x x , where n is an integer and n 2. 77.f͑x͒ ϭ ͑x ϩ 4͒2͞3 78. f ͑x͒ ϭ x2 Ϫ 4 (d) Find fЈ͑x͒ if f ͑x͒ ϭ x4. Compare the result with the conjecture in part (c). Is this a proof of your conjecture? y y Explain. 5 4 4 3 True or False? In Exercises 93–96, determine whether the 2 statement is true or false. If it is false, explain why or give an example that shows it is false. x x −4 3 4 −6 −4 −2 93. The slope of the tangent line to the differentiable function f at − 2 −3 the point ͑2, f ͑2͒͒ is f ͑2 ϩ⌬x͒ Ϫ f ͑2͒ x2 Ϫ 4, x Յ 0 . 79.f ͑x͒ ϭ Ίx Ϫ 1 80. f͑x͒ ϭ Ά ⌬x 4 Ϫ x2, x > 0 y y 94. If a function is continuous at a point, then it is differentiable at that point. 3 4 95. If a function has derivatives from both the right and the left at 2 2 a point, then it is differentiable at that point. 1 x −4 4 96. If a function is differentiable at a point, then it is continuous at x 1 2 3 4 that point. −4 97. Differentiability and Continuity Let

Graphical Reasoning In Exercises 81–84, use a graphing 1 x sin , x 0 utility to graph the function and find the x -values at which f is f ͑x͒ ϭ Ά x differentiable. 0, x ϭ 0 4x 81.f ͑x͒ ϭ Խx Ϫ 5Խ 82. f ͑x͒ ϭ and x Ϫ 3 1 83. f ͑x͒ ϭ x2͞5 x2 sin , x 0 g͑x͒ ϭ Ά x . x3 Ϫ 3x2 ϩ 3x, x Յ 1 0, x ϭ 0 84. f ͑x͒ ϭ Ά x2 Ϫ 2x, x > 1 Show that f is continuous, but not differentiable, at x ϭ 0. Ј͑ ͒ Determining Differentiability In Exercises 85–88, find Show that g is differentiable at 0, and find g 0 . if they 98. Writing Use a graphing utility to graph the two functions) 1 ؍ the derivatives from the left and from the right at x .f ͑x͒ ϭ x2 ϩ 1 and g͑x͒ ϭ ԽxԽ ϩ 1 in the same viewing window ?1 ؍ exist). Is the function differentiable at x Use the zoom and trace features to analyze the graphs near ͑ ͒ ϭ Ϫ ͑ ͒ ϭ Ί Ϫ 2 85.f x Խx 1Խ 86. f x 1 x the point ͑0, 1͒. What do you observe? Which function is ͑x Ϫ 1͒3, x Յ 1 x, x Յ 1 differentiable at this point? Write a short paragraph describing 87.f ͑x͒ ϭ Ά 88. f ͑x͒ ϭ Ά ͑x Ϫ 1͒2, x > 1 x2, x > 1 the geometric significance of differentiability at a point.

Chapter 2 25. (a) Tangent line: 27. (a) Tangent line: y 2x 2 y 12x 16 Section 2.1 (page 103) (b) 8 (b) 10 ͞ (2, 8) 1. m1 0, m2 5 2 − (−1, 4) 3. (a)–(c) f(4) f(1) − 5. m 5 y = (x 1) + f(1) = x + 1 − 4 − 1 7. m 4 55 y −3 3 − −1 4 6 29. (a) Tangent line: 31. (a) Tangent line: 5 f(4) = 5 (4, 5) 1 1 3 4 y 2 x 2 y 4 x 2 − f(4) f(1) = 3 3 6 3 (b) (b) f(1) = 2 2 (1, 2) (1, 1) −12 12 1 (−4, −5) x −1 5 654321

−1 − 9. m 3 11. f͑x͒ 0 13. f͑x͒ 10 10 33. y 2x 1 35. y 3x 2; y 3x 2 15. h͑s͒ 2 17. f͑x͒ 2x 1 19. f͑x͒ 3x2 12 3 37. y 1 x 3 1 1 2 2 21. f͑x͒ 23. f͑x͒ 39. y ͑x 1͒2 2Ίx 4 4 The slope of the graph of f is 1 for 3 all x-values. 2 f ′

x −3 −2 −1 123 −1

−2 41. y

4 f ′ The slope of the graph of f is 2 negative for x < 4, positive for x −62−4 −2 46x > 4, and 0 at x 4. −2

−4

−6

−8 43. y 2 The slope of the graph of f is

1 negative for x < 0 and positive for f ′ x > 0. The slope is undefined at x −2−1 1234 x 0.

−2 ͑ ͒ ͑ ͒ 5 45. Answers will vary. 47. g 4 5; g 4 3 Sample answer: y x y

4 3 2 1 x −42−3 −2 −1 34 −1 −2 −3 −4

49. f͑x͒ 5 3x 51. f͑x͒ x2 c 1 c 6

53. f͑x͒ 3x 2 55. y 2x 1; y 2x 9 85. The derivative from the left is 1 and the derivative from the y right is 1, so f is not differentiable at x 1. 87. The derivatives from both the right and the left are 0, so 2 f͑1͒ 0. 1 89. f is differentiable at x 2. x ͑ ͒͞Ί 2 −32−2 −1 3 91. (a) d 3Խm 1Խ m 1 −1 (b) 5 −2 f Not differentiable at m 1 −3

3 57. (a) −44 For this function, the slopes of −1 the tangent lines are always (−1, 1) (1, 1) f͑2 x͒ f͑2͒ −33distinct for different values of x. 93. False. The slope is lim . (0, 0) x→0 x −1 95. False. For example, f ͑x͒ ԽxԽ. The derivative from the left and the derivative from the right both exist but are not equal. (b) 3 97. Proof For this function, the slopes of (0, 0) (1, 1) −33the tangent lines are (−1, −1) sometimes the same.

−3

59. (a) 6

−66

−2 ͑ ͒ ͑1͒ 1 ͑ ͒ ͑ ͒ f 0 0, f 2 2, f 1 1, f 2 2 ͑1͒ 1 ͑ ͒ ͑ ͒ (b) f 2 2, f 1 1, f 2 2 (c) y 4 3 f ′ 2 1 x −4−3 −2 1234

−2 −3 −4

(d) f͑x͒ x 61. 3 ͑ ͒ Ϸ ͑ ͒ g g x f x

f −24

−1 63. f͑2͒ 4; f͑2.1͒ 3.99; f͑2͒ Ϸ 0.1 65. 6 67. 4 69. g(x͒ is not differentiable at x 0. 71. f͑x͒ is not differentiable at x 6. 73. h͑x͒ is not differentiable at x 7. 75. ͑, 3͒ ʜ ͑3, ͒ 77. ͑, 4͒ ʜ ͑4, ͒ 79. ͑1, ͒ 81. 7 83. 5

−6 6

−111

−1 −3 ͑, 5͒ ʜ ͑5, ͒ ͑, 0͒ ʜ ͑0, ͒