Chapter 2: Limits

1. Chapter 2: Limits 2.1 The Tangent & Velocity Problems

2. Tangent • A tangent to a curve is a line that touches the curve in exactly one spot. • Should have the same direction as the curve as the point of contact

3. Example 1 • Find an equation of the tangent line to the parabola y = x2at the point P(1,1). • We need to know the slope of the tangent… • But there’s only one point! • If we choose a point on the parabola that’s nearby, we could compute the slope of a secant line…

4. Example 1 cont… • Choose x ≠ 1 so that Q ≠ P • If we choose Q(1.5,2.25), we have:

5. Example 1 cont… • Check out the table: • If we choose several x’s close to 1, the closer Q is to P, the closer x is to 1, and the closer the slope is to 2 • Slope of the tangent line must be 2!

6. Example 1 cont… • Find an equation of the tangent line to the parabola y = x2at the point P(1,1).

7. Limits • We say that the slope of the tangent line is the limit of the slopes of the secant lines

8. Example 2 • The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the flash goes off). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04. (the slope of the tangent line represents the electric current flowing from the capacitor to the flash bulb in microamperes)

9. Velocity • If you watch the speedometer of a car, the needle doesn’t stay still long • Velocity of the car is not constant • The car has definite velocity at each moment • What is its instantaneous velocity??

10. Example 3 • Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds. • Galileo’s law:

11. Homework • P. 65 • 1, 5, 7