monday, sept 28, 2015mat 145. monday, sept 28, 2015mat 145
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Calculus I (MAT 145)Dr. Day Monday Sept 28, 2015
Derivative Patterns (3.1-3.4)
Assignments
Monday, Sept 28, 2015 MAT 145
MAT 145
Warm-Up: Derivative Rules
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MAT 145
Detecting Derivative Patterns
Derivatives of Trigonometric Functions
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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
Monday, Sept 28, 2015
MAT 145
Detecting Derivative Patterns
Derivative of a Product of Functions
Derivative of a Quotient of Functions
Monday, Sept 28, 2015
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.
Monday, Sept 28, 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.
Monday, Sept 28, 2015 MAT 145
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?
Monday, Sept 28, 2015
MAT 145
Derivatives of Composite
Functions (3.4)
Monday, Sept 28, 2015
The derivative of a composite function is :
The derivative of the outside functionevaluated at the inside function
multiplied bythe derivative of the inside function.
If h(x) = f (g(x)),
thenh (x) = f (g(x)) g (x)
MAT 145
Derivatives of Composite
Functions (3.4)
Monday, Sept 28, 2015
The derivativeof the outside function
evaluated atthe inside function
multiplied bythe derivative of
the inside function.
MAT 145
Derivatives of Composite
Functions (3.4)
Monday, Sept 28, 2015
MAT 145
Using Derivative Patterns
For s(t) = cos(2t):(a) Calculate s’(t) and s’’(t).
(b) Determine an equation for the line tangent to the graph of
s when t = π/8.
(c) Determine the two values of t closest to t = 0 that lead to horizontal tangent lines.
(d) Determine the smallest positive value of t for which s’(t) = 1.
(e) If s(t) represents an object’s position on the number line at time t (s in feet, t in minutes), calculate the object’s
velocity and acceleration at time t = π/12. Based on those
results, describe everything you can about the object’s movement at that instant.
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MAT 145Monday, Sept 28, 2015
THE CHAIN RULEWORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE"Here's a function in a function And your job here is to findThe derivative of the whole thingWith respect to x inside.
Call the outside f of uAnd call the inside u of x.Differentiate to find df/duAnd multiply by du/dx.
Use the chain rule.Use the chain rule.Use the chain rule whene'er you find The derivative of a function compositionally defined.
MAT 145
WebAssign
3.1 and 3.2, two parts for each, due throughout the next few days. 3.1 part 2: tonight 3.2 part 2 and 3.3: Tuesday
Assignments
Monday, Sept 28, 2015
MAT 145Monday, Sept 28, 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 145Monday, Sept 28, 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 145Monday, Sept 28, 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 145Monday, Sept 28, 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 145Monday, Sept 28, 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?
MAT 145Monday, Sept 28, 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 145Monday, Sept 28, 2015