Friday, October 23, 2015MAT 145 Please review TEST #2 Results and see me with questions

Calculus I (MAT 145)Dr. Day Friday October 23, 2015 How Can We Use Derivative Information?

Increasing/Decreasing, Maxs/Mins, Concavity, Points of Inflection, Graph the Function, Solving Optimization Problems

Assignments Using Derivatives (II) due tonight! Using Derivatives (III) due Sunday night!

Next Test: Friday, October 30, 2015

Friday, October 23, 2015 MAT 145

Please review TEST #2 Results and see me with questions.

MAT 145

Warm-Up: Mathematize This!

A rectangular field is to be fenced. If we represent the field as rectangle ABCD, the cost of the fence for side AB is $10 per foot, the cost of the fence for side CD is $10 per foot, the cost of the fence for side BC is $3 per foot, and the cost of the fence for side DA is $7 per foot.

We have $700 available to pay for fence materials.

If x represents the length of side AB, create a function A(x) to represent the area of the rectangle in terms of x.

Friday, October 23, 2015

MAT 145

Mathematize This!PROBLEM: A rectangular field is to be fenced. If we represent the field as rectangle ABCD, the cost of the fence for side AB is $10 per foot, the cost of the fence for side CD is $10 per foot, the cost of the fence for side BC is $3 per foot, and the cost of the fence for side DA is $7 per foot.



Need help?

Create a sketch to represent the situation. Label points.

Suppose side AB is 12 feet long. What does the fence cost for that side? What more can you conclude using this information?

Now let side AB measure x feet in length. What is the domain for x?


MAT 145

Mathematize This!


MAT 145

Mathematize This!


MAT 145

Mathematize This!


MAT 145

Mathematize This!


MAT 145

Mathematize This!


dynamic view on YouTube

http://youtu.be/BGeC4Oqffyc?hd=1

http://youtu.be/rnGZONQzhNg?hd=1

MAT 145

Mathematize This!


A rectangular field is to be fenced. If we represent the field as rectangle ABCD, the cost of the fence for side AB is $10 per foot, the cost of the fence for side CD is $10 per foot, the cost of the fence for side BC is $3 per foot, and the cost of the fence for side DA is $7 per foot.



MAT 145

Do Not Use a Graphing Device!


For h(x) shown below, use calculus to determine and justify:• All x-axis intervals for which h is increasing• All x-axis intervals for which h is decreasing• The location and value of every local & absolute

extreme.

• All x-axis intervals for which h is concave up• All x-axis intervals for which h is concave

down• The location of every point of inflection and

all asymptotes .

Finally, use all this info to sketch a graph of h!

MAT 145

Absolute and Relative Maximums

and MinimumsMust every continuous function have

critical points on a closed interval? Explain.

Can an increasing function have a local max? Explain.


MAT 145

First Derivative Test, Concavity, Second

Derivative Test (4.3)Determining Increasing or Decreasing Nature of a FunctionIf f’(x) > 0, then f is _?_.If f’(x) < 0, then f is _?_.

Using the First Derivative to Determine Whether an Extreme Value Exists: The First Derivative Test (and first derivative sign charts)

If f’ changes from positive to negative at x=c, then f has a _?_ _?_ at c.

If f’ changes from negative to positive at x=c, then f has a _?_ _?_ at c.

If f’ does not change sign at x=c, then f has neither a local max or min at c.

Concavity of fIf f’’(x) > 0 for all x in some interval I, then the graph is concave up on I.

If f’’(x) < 0 for all x in some interval I, then the graph is concave down on I.

Second derivative TestIf f’(c) = 0 and f’’(c) > 0, then f has a local min at c.

If f’(c) = 0 and f’’(c) < 0, then f has a local max at c.


MAT 145

Absolute and Relative Maximums

and MinimumsUse the graph of f ’(x) to describe a

strategy for identifying the global and local extrema of f, knowing f ’(x).


m a t

h

MAT 145

Info about f from f ’


Here’s a graph of g’(x). Determine all intervals over which g is increasing and over which g is decreasing. Identify and justify where all local extremes occur.

MAT 145

Info about f from f ’’


Here’s a graph of h”(x). Determine all intervals over which h is concave up and over which h is concave down. Identify and justify where all points of inflection occur.

MAT 145

Optimization! Provide complete analytical Calculus evidence to

support your responses!A row boat is exactly 7 miles from a straight shoreline. The boat operator has a vehicle 2o miles downstream from a shoreline spot perpendicular to the boat’s position. To what point on the shore should the operator aim so that she can get to her vehicle as quickly as possible? The operator can row the boat 3 mph and she can walk/run at 4 miles per hour. What is the shortest travel time possible?


MAT 145Friday, October 23, 2015












MAT 145

Derivative Patterns You Must Know 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 The Derivative of a Logarithmic Function The Derivative of a Product of Functions The Derivative of a Quotient of Functions The Derivatives of Trig Functions Derivatives of Composite Functions (Chain Rule) Implicit DifferentiationFriday, October 23, 2015

MAT 145

Position, Velocity, Acceleration


1. An object is moving in a positive direction when ….

2. An object is moving in a negative direction when ….

3. An object speeds up when ….

4. An object slows down when ….

5. An object changes directions when ….

6. The average velocity over a time interval is found by ….

7. The instantaneous velocity at a specific point in time is

found by ….

8. The net change in position over a time interval is found

by ….

9. The total distance traveled over a time interval is found

by ….

MAT 145

Position, Velocity, Acceleration


1. An object is moving in a positive direction when v(t) > 0.

2. An object is moving in a negative direction when v(t) < 0.

3. An object speeds up when v(t) and a(t) share same sign.

4. An object slows down when v(t) and a(t) have opposite signs.

5. An object changes directions when v(t) = 0 and v(t) changes sign.

6. The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!).

7. The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time.

8. The net change in position over a time interval is found by calculating the difference in the positions at the start and end of the interval.

9. The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements.

Documents

Friday, October 23, 2015MAT 145 Please review TEST #2 Results and see me with questions