wednesday, october 21, 2015mat 145 please review test #2 results and see me with questions

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Calculus I (MAT 145) Dr. Day Wednesday October 21, 2015 How Can We Use Derivative Information? Increasing/Decreasing, Maxs/Mins, Concavity, Points of Inflection, Graph the Function Assignments Using Derivatives (I) due tonight! Using Derivatives (II) due tomorrow night! Next Test : Friday, October 30, 2015 Wednesday, October 21, 2015 MAT 145 Please review TEST #2 Results and see me with questions.

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Page 1: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

Calculus I (MAT 145)Dr. Day Wednesday October 21,

2015 How Can We Use Derivative Information?

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

Assignments Using Derivatives (I) due tonight! Using Derivatives (II) due tomorrow night!

Next Test: Friday, October 30, 2015

Wednesday, October 21, 2015

MAT 145

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

Page 2: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Warm-Up: Mathematize This!

A bendable wire measures 1000 cm in length. Write a function, call it A(x), to represent the sum of the areas of a circle and a square that result when that wire is cut at one point and the resulting two pieces of wire are used to create those shapes.Wednesday, October 21, 2015

Page 3: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!PROBLEM: A bendable wire measures 1000 cm in length. Write a function, call it A(x), to represent the sum of the areas of a circle and a square that result when that wire is cut at one point and the resulting two pieces of wire are used to create those shapes.

Need help?

Create a sketch to represent the situation.

Suppose the wire is cut at a point 400 cm from one end. What are the dimensions of the resulting two shapes? Their areas?

Now suppose the wire is cut at a point x cm from one end. Repeat the previous prompt. What is the domain for x?Wednesday, October 21, 2015

Page 4: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!

Wednesday, October 21, 2015

Page 5: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!

Wednesday, October 21, 2015

Page 6: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!

Wednesday, October 21, 2015

Page 7: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!

Wednesday, October 21, 2015 dynamic view on YouTube

Page 8: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Mathematize This!

Wednesday, October 21, 2015

A bendable wire measures 1000 cm in length. Write a function, call it A(x), to represent the sum of the areas of a circle and a square that result when that wire is cut at one point and the resulting two pieces of wire are used to create those shapes.

Page 10: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Concavity of a Function

Wednesday, October 21, 2015

Concavity Animations

More Concavity Animations

Page 11: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Do Not Use a Graphing Device!

Wednesday, October 21, 2015

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

extreme

• All x-axis intervals for which g is concave up• All x-axis intervals for which g is concave down• The location of every point of inflection.

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

Page 12: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145Wednesday, October 21, 2015

Page 13: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145Wednesday, October 21, 2015

Page 14: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145Wednesday, October 21, 2015

Page 15: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

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).

Wednesday, October 21, 2015

m a t

h

Page 16: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

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.

Wednesday, October 21, 2015

Page 17: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

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.

Wednesday, October 21, 2015

Page 18: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Info about f from f ’

Wednesday, October 21, 2015

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.

Page 19: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Info about f from f ’’

Wednesday, October 21, 2015

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.

Page 20: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Optimization! Provide complete analytical Calculus evidence to

support your responses!

Two positive integers have a product of 1600. Determine the value of each integer so that the sum of the two integers is as small as possible. What is that sum?Wednesday, October 21,

2015

Page 21: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

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?

Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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MAT 145Wednesday, October 21, 2015

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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 DifferentiationWednesday, October 21, 2015

Page 35: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Position, Velocity, Acceleration

Wednesday, October 21, 2015

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 ….

Page 36: Wednesday, October 21, 2015MAT 145 Please review TEST #2 Results and see me with questions

MAT 145

Position, Velocity, Acceleration

Wednesday, October 21, 2015

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.