Wednesday, Sept 30, 2015MAT 145. Wednesday, Sept 30, 2015MAT 145

Calculus I (MAT 145)Dr. Day Wednesday Sept 30,

2015

Derivative Patterns (3.1-3.4)

Assignments

Wednesday, Sept 30, 2015

MAT 145

MAT 145

Warm-Up: Derivative Rules


MAT 145

Detecting Derivative Patterns

Derivatives of Trigonometric Functions


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.


MAT 145

MAT 145


(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?


MAT 145

Derivatives of Composite

Functions (3.4)


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


Functions (3.4)


The derivativeof the outside function

evaluated atthe inside function

multiplied bythe derivative of

the inside function.

MAT 145Wednesday, Sept 30, 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


Functions (3.4)


MAT 145


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.


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



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?



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.



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?



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.


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



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?



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.


• 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.



A rabbit moves within a straight horizontal tube according to the position function where s(t) is measured in feet and t in seconds. When s(t) is positive, the rabbit is to the right of some arbitrary point labeled 0.

4. Determine the rabbit’s velocity and acceleration functions and then sketch graphs of position, velocity, and acceleration on the axis provided on the back side, over the interval from 0 to 6 seconds.

5. Determine the times from 0 to 6 seconds when the rabbit is moving to the right and when it is moving to the left. Explain your response with justification.

6. Determine when the rabbit is speeding up and when it is slowing down. Explain your response with justification.


MAT 145


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, including whether the object is speeding up or slowing down.





A particle moves on a number line with its position given by where s(t) is measured in cm and t in seconds.

1. Determine the particle’s velocity and acceleration functions and then sketch graphs of position, velocity, and acceleration on the interval from 0 to 6 seconds.

2. Determine the times from 0 to 6 seconds when the particle is moving to the right, when it is moving to the left, and when it is at rest . Use interval notation or inequality notation. Explain and justify your response.

3. Determine when the particle is speeding up and when it is slowing down. Use interval notation or inequality notation. Explain and justify your response.

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Wednesday, Sept 30, 2015MAT 145. Wednesday, Sept 30, 2015MAT 145