Calculus I (MAT 145)Dr. Day Friday Sept 25, 2015

Derivative Patterns (3.1-3.2)

Assignments

Friday, Sept 25, 2015 MAT 145

MAT 145Friday, Sept 25, 2015

The derivative in action!

S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?

MAT 145Friday, Sept 25, 2015

The derivative in action!

S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?

• From the description of the context, the “rate units” are: feet per minute.

• The value 12 is an input variable, so we are looking at the precise instant that 12 minutes of travel has occurred, since some designated starting time when t = 0.

• S’ indicates rate of change of S, indicating we have information about how S is changing with respect to t, in feet per minute.

• The value 100 specifies the rate: 100 feet per minute.

Putting it all together:

• At precisely 12 minutes into the trip, the object’s position is increasing at the rate of 100 feet per minute.

MAT 145Friday, Sept 25, 2015

The derivative in action!

C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?

MAT 145Friday, Sept 25, 2015

The derivative in action!

C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?• From the description of the context, the “rate units” are:

thousands of dollars per patient.• The value 90 is an input variable, so we are looking at the

precise instant when 90 patients are in the hospital.• C’ indicates rate of change of C, indicating we have

information about how C is changing with respect to p, in thousands of dollars per patient.

• The value 4.5 specifies the rate: 4.5 thousand dollars ($4500) per patient.

Putting it all together:

• At precisely the instant that 90 patients are in the hospital, the cost per patient is increasing at the rate of $4500 per patient.

MAT 145Friday, Sept 25, 2015

The derivative in action!

V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π?

• From the description of the context, the “rate units” are: cubic cm of volume per cm of radius.

• The value 3 is an input variable, so we are looking at the precise instant when the sphere’s radius is 3 cm long.

• V’ indicates rate of change of V, indicating we have information about how V is changing with respect to r, in cubic cm per cm.

• The value 36π specifies the rate: 36π cubic cm of volume per 1 cm of radius length.

Putting it all together:

• At precisely the instant that the sphere has a radius length of 3 cm, the sphere’s volume is increasing at the rate of 36π cubic cm per cm of radius length.

MAT 145Friday, Sept 25, 2015

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MAT 145Friday, Sept 25, 2015

MAT 145

Detecting Derivative Patterns

The Derivative of a Constant Function

The Derivative of a Power Function

The Derivative of a Function Multiplied by a Constant

The Derivative of a Sum or Difference of Functions

The Derivative of a Polynomial Function

The Derivative of an Exponential Function

Friday, Sept 25, 2015

UsingLeibniz notation:

DERIVATIVE of a CONSTANT

If n is any real number,

then

POWER RULE

If c is a constant and f is a

differentiable

function, then

CONSTANT MULTIPLE RULE

If f and g are both differentiable,

then

SUM and DIFFERENCE RULES

The Constant Multiple

Rule, the Sum Rule, and

the Difference Rule can

be combined with the

Power Rule to

differentiate any

polynomial.

NEW DERIVATIVES FROM OLD

The derivative of the natural

exponential function is:

DERIVATIVE OF NATURAL EXPONENTIAL FUNCTION

For an exponential function

f(x) = ax with a > 0,

f’(x) = ln(a) ax

EXPONENTIAL FUNCTIONS

MAT 145

Detecting Derivative Patterns

Derivative of a Product of Functions

Derivative of a Quotient of Functions

Friday, Sept 25, 2015

If f and g are both differentiable, then:

In words, the Product Rule says:

The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

THE PRODUCT RULE

If f and g are differentiable, then:

In words, the Quotient Rule says:

The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

THE QUOTIENT RULE

MAT 145

Using Derivative Patterns

For f(x) = 2x2 – 3x + 1:

(a) Calculate f’(x).

(b) Determine an equation for the line tangent to the graph of f when x = −1.

(c) Determine all values of x that lead to a horizontal tangent line.

(d) Determine all ordered pairs of f for which f’(x) = 1.

Friday, Sept 25, 2015

Using Derivative Patterns

Suppose s(x), shown below, represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds, for x > 0.

(a) Calculate the object’s velocity and acceleration functions.

(b) Is the object moving left or right at time x = 1? Justify.

(c) Determine the object’s velocity and acceleration at time x = 2. Based on those results, describe everything you can about the object’s movement at that instant.

(d) Write an equation for the tangent line to the graph of s at time x = 1.

Friday, Sept 25, 2015

MAT 145

Using Derivative Patterns

(a)Determine the equation for the line tangent to the graph of g at x = 4.

(b)Determine the equation for the line normal to the graph of g at x = 1.

(c)At what points on the graph of g, if any, will a tangent line to the curve be parallel to the line 3x – y = –5?

Friday, Sept 25, 2015

MAT 145

WebAssign

3.1 and 3.2, two parts for each, due throughout the next few days. 3.1 part 1: tonight 3.2 part 1: Sunday

Assignments

Friday, Sept 25, 2015