Name: ____________________________ Chapter 9 – The Derivative in Calculus (UCSMP – Pre-Calculus and Discrete Mathematics) 9.1 Difference Quotients and Rates of Change

Length of day (in minutes) on first day of month at 50o north latitude Date Jan 1 Feb 1 Mar 1 Apr 1 May1 Jun 1 Jul 1 Aug 1 Sep 1 Oct 1 Nov 1 Dec 1 Day of year

1 32 60 91 121 152 182 213 244 274 305 335

Length 490 560 659 775 882 964 977 913 809 698 587 504 The change in days from x1 to x2 is x2 – x1. This is called ___________________________. The change in length of day from f(x1) to f(x2) is ________________________ and is called ____________________________. Determine how fast the length of day has been changing from April 1st to May 1st. In general, if y = f(x), then ____________________________________________________ Definition Let f be a function defined at x1 and x2 with 21 xx ≠ . The __________________________________ of a function from x1 to x2 is the slope of the line through (x1, f(x1)) and (x2, f(x2)).

1

A projectile is propelled into the air from ground level with an initial velocity of 800 ft/sec. If Earth’s gravity is considered to be a constant, its height (in feet) after t seconds is given by the function h(t) = 800t - 16t2. This function is graphed below for . Several secant lines are also drawn. Find the average rate of change of height with respect to time over the following intervals.

400 ≤≤ t

a.) b.) 2010 ≤≤ t 3020 ≤≤ t c.) 4030 ≤≤ t The rate of change of an object’s directed distance from a fixed point (or the position of an object on a line) over a time interval is called its average velocity. The Difference Quotient When many average rates of change are to be calculated for a particular function f, it helps to have a general formula. the average rate of change = f(x2) – f(x1) in f from x1 to x2 x2 – x1

We can rewrite this in terms of x, , and f. xΔ

the average rate of change =

in f from x1 to x2

2

Suppose h(t) = 960t – 16t2. Find a formula for the difference quotient giving the average rate of change of h for the interval from t to t + tΔ . Find the projectile’s average velocity from t to t + tΔ for t = 5 and

tΔ = 1

tΔ =0.5

tΔ =0.1

3

9.2 The Derivative at a Point Instantaneous velocity is the limit approached as the lengths of the time intervals approach zero. Definition Suppose an object is moving so that at each time t it is at position f(t). Then, instantaneous velocity of the object at time t = (average velocity of the object between times t and

0lim→Δt

tt Δ+ )

= ( ) ( )

ttfttf

t Δ−Δ+

→Δ 0lim

provided this limit exists and is finite. Example 1 For the projectile h(t) = 800t – 16t2 (from 9.1) what is the instantaneous velocity at time t = 5 seconds? Picturing Instantaneous Velocity

4

Example 2 The graph at the right shows the distance d (in miles) of a car from its starting point t hours after it begins a trip. Estimate the instantaneous velocity of the car at time t = 1 hour by sketching the tangent line to the graph at t = 1 and finding an approximate value for its slope. Definition The derivative of a real function f at x, denoted f’(x), is given by:

( ) ( ) ( )x

xfxxfxfx Δ

−Δ+=

→Δ 0lim'

provided this limit exists and is finite. Example 3 Let f be the function f(x) = x2 + 4 for all real numbers x. Find f’(3), the derivative of f at x = 3. Instantaneous Rates of Change are Derivatives!! Velocity is only one of many kinds of rates of change. It applies when x represents time and y = f(x) represents the position of an object in one dimension such as height or distance. If x and y are any quantities related by an equation y = f(x), then, for a particular value of x, the instantaneous rate of change of f at x is defined to be the derivative of f at x.

5

9.3 The Derivative Function

1. What is the value of the derivative of f at 2π

=x ?

2. Consider the point at x = 2. a.) Is the derivative positive, negative, or zero? b.) To determine the slope more accurately, calculate the derivative of f at x = 2. By definition,

( )x

xfx Δ

−Δ+=

→Δ

)2sin()2sin(lim2'0

To estimate this limit, evaluate the difference quotient for some small values of xΔ . Make certain your calculator is set to radian mode.

= 0.1 xΔ1.0

)2sin()1.02sin( −+ = _______

xΔ = 0.01 01.0

)2sin()01.02sin( −+ = _______

= 0.001 xΔ001.0

)2sin()001.02sin( −+ = _______

= 0.0001 xΔ0001.0

)2sin()0001.02sin( −+ = _______

6c.) Estimate f’(2)

3. Repeat part 2b with two different values of x between 0 and 2π . Fill in each the table.

Then estimate f’(x) for each of your values of x. Your first chosen x value: __________

xΔ 0.1 0.01 0.001 0.0001

xxxx

Δ−Δ+ )sin()sin(

Your estimate of f’(x): _________ ---------------------------------------------------------------------------------------------------------------- Your second chosen x value: ___________

xΔ 0.1 0.01 0.001 0.0001

xxxx

Δ−Δ+ )sin()sin(

Your estimate of f’(x): __________ 4. Complete the table below, examine for interesting results, and summarize.

Task x sin x cos x tan x f’(x)

1 2π 1

2 2 sin 2 909.≈

3 - 1st x-value

3 - 2nd x-value

Summary of results: Definition

7

Suppose that f is a function that has a derivative f’(x) at each point x in the domain of f. Then the function for all x in the domain of f is called the derivative function of f.

( ) ( )xfxf ':' →

Example 1 Suppose the height (in feet) of a projectile t seconds after launch is given by h(t) = 800t – 16t2. Find a formula for its velocity t seconds after launch. Graphing a Derivative Function

• Move along the graph from left to right • For each x-value, draw the tangent line and estimate the slope • Plot (x, f’(x))

Example 2 A function f is graphed below. Use the graph of f to estimate f’(x) when x = -5, -2, 2, and 6. Use this information to sketch a graph of f’ for values of x between -5 and 6. Finding Formulas for the Derivatives of Constant, Linear, and Quadratic Functions

8

9

Quadratic Functions f(x) = ax2 + bx + c

g(x) = 3x2 + 6x – 2 g’(x) = 6x + 6

h(x) = -4x2 – 7x + 2 h’(x) = -8x – 7

j(x) = 12x2 – 1 j’(x) = 24x

k(x) = -17x2

k’(x) = -34x

m(x) = 9x2 -11x +125 m’(x) = 18x – 11

f(x) = ax2 + bx + c

f’(x) = ___________ n(x) = 9x2 + 18x + 12

n’(x) =_____________

p(x) = -10x2 + 7x

p’(x) = _________ Linear Function f(x) = mx + b

q(x) = 4x + 5 q’(x) = 4

r(x) = ½ x – 6 r’(x) = ½

s(x) = -7x s’(x) = -7

f(x) = mx + b

f’(x) = ______ t(x) = 15x – 3

t’(x) = ______

u(x) = -9x

u’(x) = ____ Constant Function f(x) = k

v(x) = 8 v’(x) = 0

w(x) = 159 w’(x) = 0

z(x) = -17 z’(x) = 0

f(x) = k

f’(x) = ____ a(x) = 59

a’(x) = ____

b(x) = -96

b’(x) = ____ 9.4 Acceleration and Deceleration

10

Below are estimates of the world population in recent years, from the International Data Base of the U.S. Bureau of the Census.

Year World Population Average rate of change during previous 5 years

Average rate of change of average rate of change

1960 3,039,000,000 1965 3,345,000,000 61,200,000 1970 3,707,000,000 72,400,000 1975 4,086,000,000 75,800,000 680,000 1980 4,454,000,000 73,600,000 -440,000 1985 4,850,000000 79,200,000 1,120,000 1990 5,278,000,000 85,600,000 1,280,000 1995 5,687,000,000 81,800,000

Acceleration is ___________________________________________________________________________ Deceleration is ___________________________________________________________________________ The instantaneous acceleration a(t) of a projectile or other object at time t is defined to be ________________________________________________________________________________________ To compute it, take the derivative of the velocity function v at t: Example Find the instantaneous acceleration a(t) of a projectile whose position at time t is given by h(t) = 800t – 16t2. Determine the instantaneous acceleration of the projectile at t = 20 sec.

acceleration = derivative of velocity acceleration = derivative of (derivative of position)

Because acceleration is a derivative of a derivative, acceleration is said to be the second derivative of position with respect to time, written a(t) = s’’(t) Given an arbitrary function f, the derivative of f’ of f is called the first derivative of f. The second derivative of f is the derivative of the first derivative and is denoted f’’. 9.5 Using Derivatives to Analyze Graphs When the slopes of tangents to the graph of a function are positive, the function is increasing. When the slopes of tangents to the graph of a function are negative, the function is decreasing. Theorem Suppose f is a function whose derivative function f’ exists for all x in the interval a < x < b. (1) ____________________________________________________________________________________ (2) ____________________________________________________________________________________ Example 1 Given f(x) = x3 – 6x2 + 9x – 5. Find f’(x) and determine whether f(x) is increasing or decreasing on 1 < x < 3. Using Derivatives to Find Maxima or Minima

11

If the derivative at a particular point is zero, then the point may be a relative maximum or relative minimum. The point may also indicate where the graph is flat or momentarily ‘flattens out.’ Vertex of a Parabola Theorem Let a, b, and c be real numbers with 0≠a . Then the parabola that is the graph of f(x) = ax2 + bx +

c has its vertex at the point where abx

2−

= .

Example 2 What is the maximum height reached by h(t) = 800t – 16t2 at time t? Optimization Problem- a problem in which the value of one variable is sought to obtain the most optimal or desirable, value of another. Example 3 A rectangular region adjacent to a building is to be enclosed with 120 feet of fencing. What should the dimensions of the region be in order to maximize the enclosed area?

12