Chapter 3: Derivatives 3.1 Derivatives and Rate of Change.
-
Upload
eustace-hubbard -
Category
Documents
-
view
258 -
download
0
Transcript of Chapter 3: Derivatives 3.1 Derivatives and Rate of Change.
Chapter 3: Derivatives
3.1Derivatives and Rate of Change
Differential Calculus
• Study of how one quantity changes in relation to another quantity
• Central concept is the derivative– Uses the velocities and slopes of tangent lines
from chapter 2
Derivatives
• Special type of limit
• Like what we used to find the slope of a tangent line to a curve
• Or finding the instantaneous velocity of an object
• Interpreted as a rate of change
Slope of a Tangent Line
• If a curve has an equation of y = f(x), and we want to find the slope of a tangent line at some point P, then we would consider some nearby point Q and compute the slope of that “tangent line” as:
ax
afxfm
)()(
Slope of a Tangent Line (cont.)
• Then, we would pick a point Q that is even closer to P, and then closer, and closer
• The number that the slope would approach as we got closer to P (the limit) was the slope of the tangent line
Formal Definition
• The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope:
• As long as the limit exists!
ax
afxfm
ax
)()(lim
Example 1• Find an equation of the tangent line to the parabola
y = x2 at the point P(1,1).
Another definition for the slope of a tangent line…
h
afhafm
h
)()(lim
0
As x approaches a, h approaches 0 (because h = x – a)
So, the slope of the tangent line becomes the equation above!
Example 2• Find an equation of the tangent line to the
hyperbola y = 3/x at the point (3,1).
Average Velocity
• Ave velocity = displacementtime h
afhaf )()(
Instantaneous Velocity
• we take the average velocity over smaller and smaller intervals (as h approaches 0)
• Velocity v(a) is the limit of the average velocities:
h
afhafav
h
)()(lim)(
0
Instantaneous Velocity cont.
• This means that the velocity at time t = a is equal to the slope of the tangent line at P.
• Let’s reconsider the problem of the falling ball…
Example 3
• Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground.
• (a) what is the velocity of the ball after 5 seconds?• Remember the equation of motion s = f(t) = 4.9t2
Example 3 cont.
• Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground.
• (b) how fast is the ball traveling when it hits the ground?
Definition
• The derivative of a function f at a number a, denoted by f’(a), is:
• If that limit exists.
h
afhafaf
h
)()(lim)('
0
Equivalent way to write the derivative
ax
afxfaf
ax
)()(lim)('
Example 4
• Find the derivative of the function f(x) = x2 – 8x + 9 at the number a.
A note….
• The tangent line to y = f(x) at some point (a,f(a)), is the line through (a,f(a)) whose slope is equal to f’(a), the derivative of f at a.
Example 5• Find an equation of the tangent line to the parabola
y = x2 – 8x + 9 at the point (3,-6).
Rates of Change
• Suppose y is a quantity that depends on another quantity x
• This means y is a function of x, and we write y = f(x)• If x changes from x1 to x2, then the change in x
(called the increment of x) is:
• The corresponding change in y is:
12 xxx
)()( 12 xfxfy
Rates of Change (cont.)
• The difference quotient (slope) is:
• Called the average rate of change of y with respect to x
12
12 )()(
xx
xfxf
x
y
Instantaneous Rate of Change
12
12
0
)()(limlim
12 xx
xfxf
x
yxxx
Definition
• The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.
Homework
• P. 120
• 5, 7, 9 a&b, 13, 17, 25, 27, 29