1. 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

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

3. 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)3638404244 Heartbeats2530266128062948 3080

4. 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 t (min)3638404244 Heartbeats2530266128062948 3080

5. 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.

6. 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 = 58t – 0.83t 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.

7. Example The position of a car is given by the values in the table: a)Find the average velocity for the time period beginning when t = 2 and lasting 3s, 2s, 1s. t (sec)012345 s (feet)0103270119178 b)Use the graph of s(t) to estimate the instantaneous velocity (the limit of average velocities) when t = 2.

8. Definition (Limit) We write lim f(x) = L (or f(x) - > L as 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. x - > a

9. Definition (Right-hand limit) We write lim f(x) = L (or f(x) - > L as 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. x - > a +

10. Definition (Left-hand limit) We write lim f(x) = L (or f(x) - > L as 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. x - > a -

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