2 Differentiation 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 2.5 THE CHAIN RULE 2.6 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 2.7 IMPLICIT DIFFERENTIATION 2.8 THE MEAN VALUE THEOREM © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Secant Lines and Tangent Lines Consider the curve y = x2 + 1. A secant line is a line between a pair of points on the curve. The secant line shown here has a slope And the secant line has the equation © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Secant Lines and Tangent Lines Choose the second point to be much closer to (1, 2), say, (1.05, 2.1025). It is reasonable to ask, “What is the slope and the equation of the line as the second point approaches the first one?” © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Secant Lines and Tangent Lines Continue this process by computing the slope of the secant line joining (1, 2) and the unspecified point (1 + h, f (1 + h)), for some value of h close to 0 (but h = 0). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Secant Lines and Tangent Lines Notice that as h approaches 0, the slope of the secant line approaches 2, which we define to be the slope of the tangent line, or the line tangent to the curve at the point (1,2). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 2.1 Tangent lines may intersect a curve at more than one point. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY The General Case The difference quotient. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.1 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.1 Finding the Equation of a Tangent Line © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.1 Finding the Equation of a Tangent Line © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.1 Finding the Equation of a Tangent Line © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.2 Tangent Line to the Graph of a Rational Function © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 13

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.2 Tangent Line to the Graph of a Rational Function © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 14

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.2 Tangent Line to the Graph of a Rational Function © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 15

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.3 Graphical and Numerical Approximation of Tangent Lines © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.3 Graphical and Numerical Approximation of Tangent Lines A reasonable estimate of the slope of the tangent line at the point (0, −1) is then 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 17

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.3 Graphical and Numerical Approximation of Tangent Lines A reasonable estimate of the slope of the tangent line at the point (0, −1) is then 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 18

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Velocity Suppose that the function s(t) gives the position at time t of an object moving along a straight line. That is, s(t) gives the displacement (signed distance) from a fixed reference point, so that s(t) < 0 means that the object is located |s(t)| away from the reference point, but in the negative direction. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 19

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Velocity Then, for two times a and b (where a < b), s(b) − s(a) gives the signed distance between positions s(a) and s(b). The average velocity vavg is then given by © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 20

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.4 Finding Average Velocity The position of a car after t minutes driving in a straight line is given by where s is measured in miles and t is measured in minutes. Approximate the velocity at time t = 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 21

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.4 Finding Average Velocity Averaging over the 2 minutes from t = 2 to t = 4, we get © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 22

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.4 Finding Average Velocity We get an improved approximation by averaging over just one minute: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 23

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.4 Finding Average Velocity It stands to reason that, if we compute the average velocity over the time interval [2, 2 + h] (where h > 0) and then let h → 0, the resulting average velocities should be getting closer and closer to the velocity at the instant t = 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 24

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.4 Finding Average Velocity It appears that the average velocity is approaching 1 mile/minute (60 mph), as h → 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 25

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.2 If s(t) represents the position of an object relative to some fixed location at time t as the object moves along a straight line, then the instantaneous velocity at time t = a is given by provided the limit exists. The speed is the absolute value of the velocity. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 26

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.5 Finding Average and Instantaneous Velocity Suppose that the height of a falling object t seconds after being dropped from a height of 64 feet is given by s(t) = 64 − 16t2 feet. Find the instantaneous velocity at time t = 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 27

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.5 Finding Average and Instantaneous Velocity © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 28

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.5 Finding Average and Instantaneous Velocity The negative velocity indicates that the object is moving in the negative (or downward) direction. The speed of the object is 64 ft/s. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 29

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Velocity Note that the instantaneous velocity at time t = 2 corresponds to the slope of the tangent line at t = 2 on the graph of s(t) vs. t. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 30

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Rates of Change Velocity is a rate (more precisely, the instantaneous rate of change of position with respect to time). In general, the average rate of change of a function f between x = a and x = b (a ≠ b) is given by © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 31

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY Rates of Change The instantaneous rate of change of f at x = a is given by provided the limit exists. The units of the instantaneous rate of change are the units of f divided by (or “per”) the units of x. Recognize this limit as the slope of the tangent line to y = f (x) at x = a. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 32

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.7 A Graph with No Tangent Line at a Point Determine whether there is a tangent line to the graph of y = |x| at x = 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 33

TANGENT LINES AND VELOCITY 2.1 TANGENT LINES AND VELOCITY 1.7 A Graph with No Tangent Line at a Point The tangent line does not exit. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 34