Chapter 2: Limits

2.1The Tangent & Velocity Problems

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

Example 1• Find an equation of the

tangent line to the parabola y = x2 at 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…

Example 1 cont…• Choose x ≠ 1 so that Q ≠ P

• If we choose Q(1.5,2.25), we have:

1

12

x

xmPQ

5.25.0

25.1

15.1

125.2

PQm

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!

X mPQ

2 3

1.5 2.5

1.1 2.1

1.01 2.01

1..001 2.001

Example 1 cont…• Find an equation of the

tangent line to the parabola y = x2 at the point P(1,1).

Limits

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

mmPQPQ

lim 2

1

1lim

2

1

x

xx

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)

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??

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: 29.4)( tts

Homework

• P. 65

• 1, 5, 7