2 The Derivative. 2.0 Introduction to Derivatives. Slopes of Tangent Lines: Graphically

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Daisy Sanders
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1 2 Te Derivative Te two previous capters ave laid te foundation for te study of calculus. Tey provided a review of some material you will need and started to empasize te various ways we will view and use functions: functions given by graps, equations and tables of values. Capter 2 will focus on te idea of tangent lines. We will develop a definition for te derivative of a function and calculate derivatives of some functions using tis definition. Ten we will examine some of te properties of derivatives, see some relatively easy ways to calculate te derivatives, and begin to look at some ways we can use tem. 2.0 Introduction to Derivatives Tis section begins wit a very grapical approac to slopes of tangent lines. It ten examines te problem of finding te slopes of te tangent lines for a single function, y = x 2, in some detail and illustrates ow tese slopes can elp us solve fairly sopisticated problems. Slopes of Tangent Lines: Grapically Te figure in te margin sows te grap of a function y = f (x). We can use te information in te grap to fill in te table: x y = f (x) m(x) were m(x) is te (estimated) slope of te line tangent to te grap of y = f (x) at te point (x, y).

2 110 te derivative We can estimate te values of m(x) at some non-integer values of x as well: m(0.5) 0.5 and m(1.3) 0.3, for example. We can even say someting about te beavior of m(x) over entire intervals: if 0 < x < 1, ten m(x) is positive, for example. Te values of m(x) definitely depend on te values of x (te slope varies as x varies, and tere is at most one slope associated wit eac value of x) so m(x) is a function of x. We can use te results in te table to elp sketc te grap of te function m(x) (see te margin). Practice 1. Te grap of y = f (x) appears in te margin. Set up a table of (estimated) values for x and m(x), te slope of te line tangent to te grap of y = f (x) at te point (x, y), and ten sketc a grap of te function m(x). In some applications, we need to know were te grap of a function f (x) as orizontal tangent lines (tat is, were te slope of te tangent line = 0). In te margin figure, te slopes of te tangent lines to grap of y = f (x) are 0 wen x = 2 or x Practice 2. At wat values of x does te grap of y = g(x) (in te margin) ave orizontal tangent lines? Example 1. Te grap of te eigt of a rocket at time t appears in te margin. Sketc a grap of te velocity of te rocket at time t. (Remember from our previous work tat instantaneous velocity corresponds to te slope of te line tangent to te grap of position or eigt function.) Solution. Te last margin grap sows te velocity of te rocket. Practice 3. Te grap below sows te temperature during a summer day in Cicago. Sketc a grap of te rate at wic te temperature is canging at eac moment in time. (As wit instantaneous velocity, te instantaneous rate of cange for te temperature corresponds to te slope of te line tangent to te temperature grap.) Te function m(x), te slope of te line tangent to te grap of y = f (x) at (x, f (x)), is called te derivative of f (x).

3 2.0 introduction to derivatives 111 We used te idea of te slope of te tangent line all trougout Capter 1. In Section 2.1, we will formally define te derivative of a function and begin to examine some of its properties, but first let s see wat we can do wen we ave a formula for f (x). Tangents to y = x 2 Wen we ave a formula for a function, we can determine te slope of te tangent line at a point (x, f (x)) by calculating te slope of te secant line troug te points (x, f (x)) and (x +, f (x + )): msec = f (x + ) f (x) (x + ) (x) and ten taking te limit of msec as approaces 0: m tan msec f (x + ) f (x) (x + ) (x) Example 2. Find te slope of te line tangent to te grap of y = f (x) = x 2 at te point (2, 4). Solution. In tis example, x = 2, so x + = 2 + and f (x + ) = f (2 + ) = (2 + ) 2. Te slope of te tangent line at (2, 4) is m tan msec f (2 + ) f (2) (2 + ) (2) (2 + ) = 4 Te line tangent to y = x 2 at te point (2, 4) as slope 4. We can use te point-slope formula for a line to find an equation of tis tangent line: y y 0 = m(x x 0 ) y 4 = 4(x 2) y = 4x 4 Practice 4. Use te metod of Example 2 to sow tat te slope of te line tangent to te grap of y = f (x) = x 2 at te point (1, 1) is m tan = 2. Also find te values of m tan at (0, 0) and ( 1, 1). It is possible to compute te slopes of te tangent lines one point at a time, as we ave been doing, but tat is not very efficient. You sould ave noticed in Practice 4 tat te algebra for eac point was very similar, so let s do all te work just once, for an arbitrary point (x, f (x)) = (x, x 2 ) and ten use te general result to find te slopes at te particular points we re interested in.

4 112 te derivative Te slope of te line tangent to te grap of y = f (x) = x 2 at te arbitrary point (x, x 2 ) is: m tan msec (x + ) 2 x 2 2x + 2 f (x + ) f (x) (x + ) (x) x 2 + 2x + 2 x 2 2x + = 2x Te slope of te line tangent to te grap of y = f (x) = x 2 at te point (x, x 2 ) is m tan = 2x. We can use tis general result at any value of x witout going troug all of te calculations again. Te slope of te line tangent to y = f (x) = x 2 at te point (4, 16) is m tan = 2(4) = 8 and te slope at (p, p 2 ) is m tan = 2(p) = 2p. Te value of x determines te location of our point on te curve, (x, x 2 ), as well as te slope of te line tangent to te curve at tat point, m tan = 2x. Te slope m tan = 2x is a function of x and is called te derivative of y = x 2. Simply knowing tat te slope of te line tangent to te grap of y = x 2 is m tan = 2x at a point (x, y) can elp us quickly find an equation of te line tangent to te grap of y = x 2 at any point and answer a number of difficult-sounding questions. Example 3. Find equations of te lines tangent to y = x 2 at te points (3, 9) and (p, p 2 ). Solution. At (3, 9), te slope of te tangent line is 2x = 2(3) = 6, and te equation of te line is y 9 = 6(x 3) y = 6x 9. At (p, p 2 ), te slope of te tangent line is 2x = 2(p) = 2p, and te equation of te line is y p 2 = 2p(x p) y = 2px p 2. Example 4. A rocket as been programmed to follow te pat y = x 2 in space (from left to rigt along te curve, as seen in te margin figure), but an emergency as arisen and te crew must return to teir base, wic is located at coordinates (3, 5). At wat point on te pat y = x 2 sould te captain turn off te engines so tat te sip will coast along a pat tangent to te curve to return to te base? Solution. You migt spend a few minutes trying to solve tis problem witout using te relation m tan = 2x, but te problem is muc easier if we do use tat result. Let s assume tat te captain turns off te engine at te point (p, q) on te curve y = x 2 and ten try to determine wat values p and q must ave so tat te resulting tangent line to te curve will go troug te point (3, 5). Te point (p, q) is on te curve y = x 2, so q = p 2 and te equation of te tangent line, found in Example 3, must ten be y = 2px p 2.

5 2.0 introduction to derivatives 113 To find te value of p so tat te tangent line will go troug te point (3, 5), we can substitute te values x = 3 and y = 5 into te equation of te tangent line and solve for p: y = 2px p 2 5 = 2p(3) p 2 p 2 6p + 5 = 0 (p 1)(p 5) = 0 Te only solutions are p = 1 and p = 5, so te only possible points are (1, 1) and (5, 25). You can verify tat te tangent lines to y = x 2 at (1, 1) and (5, 25) bot go troug te point (3, 5). Because te sip is moving from left to rigt along te curve, te captain sould turn off te engines at te point (1, 1). (Wy not at (5, 25)?) Practice 5. Verify tat if te rocket engines in Example 4 are sut off at (2, 4), ten te rocket will go troug te point (3, 8). 2.0 Problems 1. Use te function f (x) graped below to fill in te table and ten grap m(x), te estimated slope of te tangent line to y = f (x) at te point (x, y). x f (x) m(x) x f (x) m(x) Use te function g(x) graped below to fill in te table and ten grap m(x), te estimated slope of te tangent line to y = g(x) at te point (x, y). x g(x) m(x) x g(x) m(x)

6 114 te derivative 3. (a) At wat values of x does te grap of f (sown below) ave a orizontal tangent line? (b) At wat value(s) of x is te value of f te largest? Smallest? (c) Sketc a grap of m(x), te slope of te line tangent to te grap of f at te point (x, f (x)). 6. Matc te situation descriptions wit te corresponding time-velocity graps sown below. (a) A car quickly leaving from a stop sign. (b) A car sedately leaving from a stop sign. (c) A student bouncing on a trampoline. (d) A ball trown straigt up. (e) A student confidently striding across campus to take a calculus test. (f) An unprepared student walking across campus to take a calculus test. 4. (a) At wat values of x does te grap of g (sown below) ave a orizontal tangent line? (b) At wat value(s) of x is te value of g te largest? Smallest? (c) Sketc a grap of m(x), te slope of te line tangent to te grap of g at te point (x, g(x)). Problems 7 10 assume tat a rocket is following te pat y = x 2, from left to rigt. 7. At wat point sould te engine be turned off in order to coast along te tangent line to a base at (5, 16)? 8. At (3, 7)? 9. At (1, 3)? 10. Wic points in te plane can not be reaced by te rocket? Wy not? In problems 11 16, perform tese steps: (a) Calculate and simplify: 5. (a) Sketc te grap of f (x) = sin(x) on te interval 3 x 10. (b) Sketc a grap of m(x), te slope of te line tangent to te grap of sin(x) at te point (x, sin(x)). (c) Your grap in part (b) sould look familiar. Wat function is it? msec = f (x + ) f (x) (x + ) (x) (b) Determine m tan msec. (c) Evaluate m tan at x = 2. (d) Find an equation of te line tangent to te grap of f at (2, f (2)). 11. f (x) = 3x f (x) = 2 7x

7 2.0 introduction to derivatives f (x) = ax + b were a and b are constants 14. f (x) = x 2 + 3x 15. f (x) = 8 3x f (x) = ax 2 + bx + c were a, b and c are constants In problems 17 18, use te result: f (x) = ax 2 + bx + c m tan = 2ax + b 17. Given f (x) = x 2 + 2x, at wic point(s) (p, f (p)) does te line tangent to te grap at tat point also go troug te point (3, 6)? 18. (a) If a = 0, ten wat is te sape of te grap of y = f (x) = ax 2 + bx + c? (b) At wat value(s) of x is te line tangent to te grap of f (x) orizontal? 2.0 Practice Answers 1. Approximate values of m(x) appear in te table in te margin; te margin figure sows a grap of m(x). x f (x) m(x) 2. Te tangent lines to te grap of g are orizontal (slope = 0) wen x 1, 1, 2.5 and Te figure below sows a grap of te approximate rate of temperature cange (slope) At (1, 1), te slope of te tangent line is: m tan msec f (1 + ) f (1) (1 + ) (1) (1 + ) = 2

8 116 te derivative so te line tangent to y = x 2 at te point (1, 1) as slope 2. At (0, 0): m tan msec f (0 + ) f (1) (0 + ) (0) (0 + ) = 2 lim = 0 so te line tangent to y = x 2 at te point (0, 0) as slope 0. At ( 1, 1): m tan msec ( 1 + ) 2 ( 1) f ( 1 + ) f ( 1) ( 1 + ) ( 1) = 2 so te line tangent to y = x 2 at te point ( 1, 1) as slope From Example 4 we know te slope of te tangent line is m tan = 2x, so te slope of te tangent line at (2, 4) is m tan = 2x = 2(2) = 4. Te tangent line as slope 4 and goes troug te point (2, 4), so an equation of te tangent line (using y y 0 = m(x x 0 )) is y 4 = 4(x 2) or y = 4x 4. Te point (3, 8) satisfies te equation y = 4x 4, so te point (3, 8) lies on te tangent line.

4.1 Tangent Lines. y 2 y 1 = y 2 y 1

4.1 Tangent Lines. y 2 y 1 = y 2 y 1 41 Tangent Lines Introduction Recall tat te slope of a line tells us ow fast te line rises or falls Given distinct points (x 1, y 1 ) and (x 2, y 2 ), te slope of te line troug tese two points is cange