The Tangent and Velocity Problems The Tangent Problem

The Tangent and Velocity Problems The Tangent Problem Can a tangent to a curve be a line that touches the curve at one point? animation

Example Guess an equation of the tangent line to the exponential function y = 2 x at the point P(0, 1). animation

Example A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table is graphed, the slope of the tangent line represents the heart rate in beats per minute. t (min) 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080

t (min) 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 Use the data to estimate the patient’heart rate after 42 minutes using the secant line between a )t = 36 and t = 42 b) t = 38 and t = 42 c) t = 40 and t = 42 d) t = 40 and t = 44

The Velocity Problem When you watch the speedometer of a car as you travel in the city traffic, you see that the needle doesn’t stay still for very long which means the velocity of the car isn’t constant.

Example If an arrow is shot upward on the moon with a velocity of 58 m/s, its height in meters after t seconds is given by h = 58 t – 0. 83 t 2. a) Find the average velocity over the given time intervals: [1, 2] [1, 1. 1] [1, 1. 5] b) Find the instantaneous velocity after 1 second.

Example The position of a car is given by the values in the table: t (sec) 0 1 2 3 4 5 s (feet) 0 10 32 70 119 178 a) Find the average velocity for the time period beginning when t = 2 and lasting 3 s, 2 s, 1 s. b)Use the graph of s(t) to estimate the instantaneous velocity (the limit of average velocities) when t = 2.

Definition (Limit) We write lim f(x) = L (or f(x) ->L as x ->a) x - >a if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a but not equal to a.

Definition (Right-hand limit) We write lim+ f(x) = L (or f(x) ->L as x ->a+ ) x - >a if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a and larger than a.

Definition (Left-hand limit) We write lim - f(x) = L (or f(x) -> L as x -> a- ) x -> a if we can make the values of f(x), arbitrarily close to L by taking x to be sufficiently close to a and less than a.

Observation If lim -f(x) = L x - >a then and lim+ f(x) = L , x - >a lim f(x) = L. x - >a