Download - Chapter 2: Limits
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