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Implicit Differentiation Related Rates Optimization Extrema Concavity Curve sketching Mean Value & Rolle’s Theorem Intermediate Value Theorem Linearization.

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Presentation on theme: "Implicit Differentiation Related Rates Optimization Extrema Concavity Curve sketching Mean Value & Rolle’s Theorem Intermediate Value Theorem Linearization."— Presentation transcript:

1 Implicit Differentiation Related Rates Optimization Extrema Concavity Curve sketching Mean Value & Rolle’s Theorem Intermediate Value Theorem Linearization & Newton’s Method TEST TOPICS

2 A student wishes to approximate by using Newton’s Method to approximate a zero of f(x) = x 2 – 12. If she chooses c 1 = 4, then c 3 will be ? Give your answer in fraction form.

3 On the same side of a straight river are two towns and the townspeople want to build a pumping station, S, that supplies water to them. The pumping station is to be at the rivers edge with pipes extending straight to the two towns. The distances are shown in the figure. Where should the pumping station be placed to minimize the total length of pipe? x = 4/5 miles

4 Sand is pouring from a pipe at the rate of 16 cubic feet per second. If the falling sand forms a conical pile on the ground whose altitude is always ¼ the diameter of the base, how fast is the altitude increasing when the pile is 4 feet high?

5 Find Consider the curve given by xy 2 – x 3 y = 6. Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent line at each of these points. Find the x-coordinate of each point on the curve where the tangent line is vertical.

6 A spherical balloon is inflating at a rate of 27π in 3 /sec. How fast is the radius of the balloon increasing when the radius is 3in.? How fast is the surface area of the balloon increasing when the radius is 3in.?

7 If an open can with volume 16  cubic inches is to be made in the form of a right circular cylinder, find the height and the radius if the least amount of material is to be used in its manufacture.

8 NON Calculator Determine (and classify) absolute extrema in the interval [0, π] for y = sin x + cos x SHOW ALL WORK.

9 Non calculator If f ”(x) = x(x +1)(x – 2) 2, then the graph of f has inflection points when x = _____ JUSTIFY YOUR ANSWER

10 Non calculator Apply the mean value theorem or state why it doesn’t apply. on the interval [0,2]

11 If has a point of inflection at x = -1, then the value of k is ___.

12 CALCULATOR ACTIVE On what parts of the interval [-2,2] is the graph of concave up? (Round to 3 decimal places)

13 Let f(x) = ln(x 2 – 3) a)Find the linearization of f(x) at x = 2. b)Use the linearization found in part (a) to estimate f(2.3). c)Determine the error of the estimate you found in part b. Y = 4x


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