Exam 3 Review. Related Rates Optimization Derivative Tests/Extrema Inverse Trig Linearization L Hopital s Rule

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1 Math 165 SI Exam 3 Review SI Leader: Riley Related Rates Optimization Derivative Tests/Extrema Inverse Trig Linearization L Hopital s Rule

2 Introductions Introduce yourself to the people around you: Name Major Favorite TV Show/Movie

3 Related Rates Draw a Picture Define variables with words Implicitly derive equation with respect to time Give answer in complete sentence

4 Related Rates Draw a Picture Define variables with words Implicitly derive equation with respect to time Give answer in complete sentence The radius of a right cylinder increases at the rate of 0.1 cm/sec, and the height decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder, in cm 3 /sec, when the radius is 2 cm and the volume is 12 cm 3?

5 Optimization Draw picture Define variables with words Write equation for value to be optimized with respect to a SINGLE variable Domain: what s the smallest/largest that variable could be? PROVE that your critical point gives a max/min (First & Second Deriv. Test, Extreme Value Thm.) Give answer in complete sentence

6 Optimization Draw picture Define variables with words Write equation for value to be optimized with respect to a SINGLE variable Domain: what s the smallest/largest that variable could be? PROVE that your critical point gives a max/min (First & Second Deriv. Test, Extreme Value Thm.) Give answer in complete sentence (F15 3A) The function f(x) = 27 - x 3 is graphed below. A student wants to approximate the area bounded by the curve, the x-axis, and the y-axis. To do so, she inscribes a rectangle in the bottom portion of the graph and a triangle in the top portion of the graph, as shown in the figure. Where should x be placed to maximize the total area of the rectangle and the triangle?

7 Optimization Draw picture Define variables with words Write equation for value to be optimized with respect to a SINGLE variable Domain: what s the smallest/largest that variable could be? PROVE that your critical point gives a max/min (First & Second Deriv. Test, Extreme Value Thm.) Give answer in complete sentence (F15 3A) The function f(x) = 27 - x 3 is graphed below. A student wants to approximate the area bounded by the curve, the x-axis, and the y-axis. To do so, she inscribes a rectangle in the bottom portion of the graph and a triangle in the top portion of the graph, as shown in the figure. Where should x be placed to maximize the total area of the rectangle and the triangle? Ans:

8 Derivative Tests/ Extrema Critical Point Definition What does the first derivative tell us about the function? What does the second derivative tell us about the function? Inflection Point Definition Where should we look for max and mins?

9 Derivative Tests/ Extrema Critical Point Definition What does the first derivative tell us about the function? What does the second derivative tell us about the function? Inflection Point Definition Where should we look for max and mins? (S17 3A) The following is the DERIVATIVE of Q(x): Identify the following information about Q: a) Critical Points b) Intervals of increasing and decreasing c) Locations of any max/mins d) Intervals of concave up and concave down e) Inflection Points

10 Derivative Tests/ Extrema Critical Point Definition What does the first derivative tell us about the function? What does the second derivative tell us about the function? Inflection Point Definition Where should we look for max and mins? (S17 3A) The following is the DERIVATIVE of Q(x): Identify the following information about Q: ANS: a) Critical Points x = 6, - 6 b) Intervals of increasing and decreasing INCR: (-inf, - 6), ( 6, inf) DECR: (- 6, 6) c) Locations of any max/mins MAX: x = - 6 MIN: x = 6 d) Intervals of concave up and concave down UP: (-inf, -3), (2, inf) DOWN: (-3, 2) e) Inflection Points x = - 3, 2

11 Inverse Trig Inverse Trig Definitions/relation to original Domains and ranges of arcsin, arccos, arctan Derivatives of arcsin, arccos, arctan Be able to rewrite expressions without trig

12 Inverse Trig Inverse Trig Definitions/Relation to original Domains and ranges of arcsin, arccos, arctan Derivatives of arcsin, arccos, arctan Be able to rewrite expressions without trig 1. State the domain and range of the function g(t) = cos ( arcsin (2t) ) 2. Simplify h(x) = sin ( arctan (2x) ) so that there are no trig or inverse trig functions. (Hint: use a right triangle)

13 Inverse Trig Inverse Trig Definitions/Relation to original Domains and ranges of arcsin, arccos, arctan Derivatives of arcsin, arccos, arctan Be able to rewrite expressions without trig 1. State the domain and range of the function g(t) = cos ( arcsin (2t) ) ANS: Domain: -½ t ½ Range: [ 0, 1 ] 2. Simplify h(x) = sin ( arctan (2x) ) so that there are no trig or inverse trig functions. (Hint: Use a right triangle) ANS:

14 L Hopital s Rule What indeterminate forms can you apply LHR to? Be able to manipulate expressions into the appropriate forms

15 L Hopital s Rule What indeterminate forms can you apply LHR to? Be able to manipulate expressions into the appropriate forms 1. 2.

16 L Hopital s Rule What indeterminate forms can you apply LHR to? Be able to manipulate expressions into the appropriate forms 1.. ANS: ANS: 3

17 Linearization Linearization Equation a the nice value x value you want to estimate in function

18 Linearization Linearization Equation a the nice value x value you want to estimate in function Use linearization to estimate arcsin ( 0.05 ) (Hint: What is the function? What is the nice value?)

19 Linearization Linearization Equation a the nice value x value you want to estimate in function Use linearization to estimate arcsin ( 0.05 ) (Hint: What is the function? What is a nice value?) ANS: 0.05

20 You got this!

The following information is for reviewing the material since Exam 3:

The following information is for reviewing the material since Exam 3: Outcomes List for Math 121 Calculus I Fall 2010-2011 General Information: The purpose of this Outcomes List is to give you a concrete summary of the material you should know, and the skills you should