Rates of Change, Slope, and Derivatives MATH 151 Calculus for Management

J. Robert Buchanan Department of Mathematics

Spring 2014

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Objectives

After completing this lesson, we will be able to: Determine the average rate of change. Determine the instantaneous rate of change. Interpret the graph of a function.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Average Rate of Change

Definition If y = f (x) then the ratio y2 − y1 f (x2 ) − f (x1 ) ∆y = = ∆x x2 − x1 x2 − x1 is called the average rate of change of y with respect to x as x changes from x1 to x2 .

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example

Find the average rate of change of f (x) = −x 2 + 1 as x changes from −1 to 3.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example

Find the average rate of change of f (x) = −x 2 + 1 as x changes from −1 to 3.

f (−1) = −(−1)2 + 1 = 0 f (3) = −32 + 1 = −8 ∆y

= f (3) − f (−1) = −8 − 0 = −8

∆x ∆y ∆x

= 3 − (−1) = 4 −8 = = −2 4

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example (1 of 2) The line passing through (1, f (1)) and the (1 + h, f (1 + h)) with h 6= 0 is called a secant line. y

2 1 x 0.5

1.0

1.5

2.0

-1

-2

-3

The slope of the secant line is the average rate of change. J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example (2 of 2)

msec = = = = = =

f (1 + h) − f (1) (1 + h) − 1 f (1 + h) − f (1) h −(1 + h)2 + 1 − (−(1)2 + 1) h −(1 + 2h + h2 ) + 1 − 0 h −2h − h2 h −2 − h

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Difference Quotient

Definition If (x, f (x)) is any fixed point on a curve and (x + h, f (x + h)) is another point on the curve, then the difference quotient is the slope of the secant line through these two points. msec =

f (x + h) − f (x) f (x + h) − f (x) = . x +h−x h

Remark: the difference quotient gives us the average rate of change of the function f between x and x + h.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Slope of a Point on a Curve

Definition The slope of a curve y = f (x) at the point (x, f (x)) is exactly the slope of the tangent line to the curve at that point. This is also called the instantaneous rate of change.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Slope of a Point on a Curve

Definition The slope of a curve y = f (x) at the point (x, f (x)) is exactly the slope of the tangent line to the curve at that point. This is also called the instantaneous rate of change. The slope of the tangent line can be found by evaluating the limit of the difference quotient as h → 0. f (x + h) − f (x) = f 0 (x) h h→0

mtan = lim

The symbol f 0 (x) is called the derivative. If the derivative exists at x, we say f is differentiable at x.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Notation for the Derivative

There are competing notations for the derivative (but they all mean the same thing). Suppose y = f (x) is a differentiable function at x. Prime Notation: f 0 (x) and y 0 denote derivative. df dy d f (x) and and denote the derivative. Leibniz Notation: dx dx dx

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example (1 of 2) Consider the function f (x) = −x 2 + 1 and the point on the graph of f with coordinates (1, f (1)). Find the slope of the tangent line to the graph at this point. y 2

1

x 0.5

1.0

1.5

2.0

-1

-2

-3

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example (2 of 2) The slope of the tangent line will be mtan = = = = =

f (1 + h) − f (1) h h→0 −(1 + h)2 + 1 − (−12 + 1) lim h h→0 2 −1 − 2h − h + 1 − (−1) − 1 lim h h→0 2 −2h − h lim h h→0 lim (−2 − h) lim

h→0

0

f (1) = −2.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Compare f (x) and f 0 (x) f HxL 3 2 1

-2

-1

1

2

x

-1

Compare the slope of the graph of f (x) at several different points with the sign of f 0 (x).

-2 -3 Out[4]=

f 'HxL -2

-1

1

2

x

-2 -4 -6 -8 -10

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Velocity One of the applications of the slope of tangent line is to determining the velocity of an object whose position is described by a function f (t). Definition If the position of an object is given by function f (t) the average velocity of the object between t = a and t = b is vavg =

f (b) − f (a) . b−a

The instantaneous velocity of the object at t = a is f (a + h) − f (a) . h h→0

v (a) = lim

The absolute value of velocity is called speed. J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Example

Suppose the height of an arrow shot into the air is given by the function s(t) = −16t 2 + 80t. Find the height of the arrow at t = 2. the average velocity of the object on the interval [1, 3]. the instantaneous velocity of the object at t = 3. the time it takes the arrow to reach its peak.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Solutions (1 of 3)

Height of the arrow at t = 2: s(2) = −16(22 ) + 80(2) = 96

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Solutions (1 of 3)

Height of the arrow at t = 2: s(2) = −16(22 ) + 80(2) = 96 Average velocity of the object on the interval [1, 3]: vavg =

96 − 64 s(3) − s(1) = = 16 ft/s 3−1 2

J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Solutions (2 of 3) Instantaneous velocity of the object at t = 3: s0 (3) = = = = = =

s(3 + h) − s(3) h h→0 −16(3 + h)2 + 80(3 + h) − 96 lim h h→0 −16(9 + 6h + h2 ) + 240 + 80h − 96 lim h h→0 −144 − 96h − 16h2 + 240 + 80h − 96 lim h h→0 2 −16h + 16h lim h h→0 lim (−16 + 16h) lim

h→0

= −16 ft/s J. Robert Buchanan

Rates of Change, Slope, and Derivatives

Solutions (3 of 3) Time it takes the arrow to reach its peak: The arrow has reached its peak when s0 (t) = 0. s(t + h) − s(t) h h→0 −16(t + h)2 + 80(t + h) − (−16t 2 + 80t) = lim h h→0 2 2 −16(t + 2th + h ) + 80t + 80h + 16t 2 − 80t = lim h h→0 −32th − 16h2 + 80h = lim h h→0 = lim (−32t − 16h + 80) = −32t + 80

s0 (t) =

lim

h→0

Thus the time at the peak is 0 = s0 (t) = −32t + 80 J. Robert Buchanan

=⇒

t = 2.5 s.

Rates of Change, Slope, and Derivatives

Summary

For a function f (x) defined at x = a, the limit of the difference quotient f (a + h) − f (a) lim = f 0 (a) h h→0 (provided the limit exists) can be interpreted as: 1

the derivative of f at x = a,

2

the instantaneous rate of change of f at x = a,

3

the slope of the tangent line to the graph of f at x = a.

J. Robert Buchanan

Rates of Change, Slope, and Derivatives