2 72 8 TANGENT LINES DERIVATIVES TANGENT LINE

2. 7/2. 8 TANGENT LINES & DERIVATIVES

TANGENT LINE The line through P = (a, f(a)) with a slope of,

DERIVATIVE The derivative of f at a is f (a):

EX 1: a) Find the slope of the tangent line at x = 4. b) Find the slope of the tangent line at x = a.

EX 2: FIND F (X) AND USE IT TO FIND THE EQUATION OF THE TANGENT LINE TO F (X) ATX = 9.

Average Velocity: the slope of the secant line between two ! t points, P & Q, on a position e g r curve. o D n o F ’t

Instantaneous Velocity: the slope of the tangent line to a ! t point P on a position curve. e g D n o F ’t r o

2. 7 PG. 154 # 1, 2, 3, 5, 11, 13, 15, 25

q NOTE: § f (x) = the slope of the tangent line at x § s (a) = the velocity at t = a § | s (a)| = the speed at t = a

EX 3: THE DISPLACEMENT (IN METERS) OF A PARTICLE MOVING IN A STRAIGHT LINE IS GIVEN WHERE T IS SECONDS.

A) FIND THE AVERAGE VELOCITY ON THE INTERVAL [4, 4. 5] B) FIND THE INSTANTANEOUS VELOCITY ATT = 4 C) FIND THE SPEED ATT = 5

EX 4: Let P(t) be the population of the United States at time t, in years. a) What are the units of P (t)? b) What does P (1992) = 2. 7 million mean?

EX 5: Sketch the graph of the function g for which g(0) = 0, g (0) = 3, g (1) = 0, & g (2) = 1.

EX 6: The limit represents the derivative of f at some number a. State f and a.

EX 7: The limit represents the derivative of f at some number a. State f and a.

2. 8 PG. 161 # 1, 3, 4, 5, 13, 15, 19 – 24 ALL, 25, 27, 28, 31