Geometry name



name ___________________

AP Calculus BC

Madison West High School

Mr. Dominic

What do you remember from Calculus AB?

I. Trigonometry

II. Sequences & Series

III. Integration Basics

IV. Polar Coordinates

V. Volumes of Solids

VI. A few proofs

VII. Student Information Page

I hope you saved all your notes from Calculus AB! They are often helpful, and you’re still responsible for knowing all of Calculus AB material.

This packet is to determine how much you remember from calculus AB. Please do NOT spend hours and hours reviewing your notes or textbook chapters. If you know how to do a problem, do it. If you only remember how to do part of a problem, do part of it. If you do not remember how to do a problem, leave it blank. I will collect this assignment at the end of class on Friday, September 7, 2007. Students’ performances on this assignment will determine how much class time we spend reviewing Calculus AB material. It will count as a homework assignment.

Standards for written work:

• 8½ in. x 11in. graph paper, with any perforations removed. Paper should be three-hole-punched to put into your class binder.

• If a graph is required, label axes and scale, if different from y vs. x and 1 unit/gridline.

• Use a black pencil only. Get a good eraser.

• The upper right-hand corner of every page must contain:

▪ Name (first and last)

▪ Class & Period

▪ Assignment name

• Show all steps leading to your solution.

• Start each problem on a new line. Feel free to use the back of the page.

• Use a straightedge or Geometer template for all sketches & graphs

• Colored pencils and/or pens are helpful for correcting your work in class.

• Late work is not accepted.

Working with friends on this packet is encouraged! Several problems require more mathematical insight than brute force methods, but if the latter is the only path to a solution that you see, then take it. If you have questions about policies or the mathematics contained herein, email me at djohannberke@madison.k12.wi.us. Enjoy the math!

I. Trigonometry Review. You should complete these without a calculator.

The only identities you need are these:

|Pythagorean |Addition |Power-reducing |

|[pic] |[pic] |[pic] |

There are many other identities, but they can be derived when needed from those above. (Actually, the third column is also superfluous, but the power-reducing identities are so common that they are worth memorizing.) Use only the above identities to solve the problems below.

1. Evaluate.

a) [pic]

b) [pic]

2. Graph and indicate scale.

a) [pic].

b) [pic]

3. Evaluate exactly and fully simplify.

a) [pic]

b) [pic]

II. Sequences and Series. Use a calculator at will. Read your calculator manual to discover how to do sum( ) and seq( ).

A sequence is an indexed set of numbers, such as {1,2,3,4, . . .}, or {3,5,8,12, . . .}.

There are two common ways of defining a sequence:

4. Each term is given by a function defined only on nonnegative integers:

[pic]

5. Recursively, where each term is generated from the previous term:

[pic]

A series is a sum, which usually evaluates to a number. Series are notated with a capital sigma in this way:

[pic].

In the above discussion, n is called the index, and takes on all integer values from start to end.

For example:

A sequence: [pic] is [pic]

A series: [pic]

Recall that 0! = 1, 1! = 1, and n! = 1 · 2 · 3 · · · n, read “n factorial.”

Problems.

6. Write the first 5 terms of the sequence [pic]

7. Write the first 5 terms of the sequence [pic]

8. Write the first 5 terms of the sequence [pic]

9. Write the first 7 terms of the sequence [pic] (look familiar?)

10. Write the general term of the sequence. [pic]

(Assume the sequence begins at n = 1 and find the formula for the nth element.)

11. Write the general term of the sequence. [pic]

12. Write the general term of the sequence. [pic]

13. Evaluate. [pic]

14. Evaluate. [pic]

15. Evaluate. [pic]

16. Evaluate. [pic]. Use your calculator or your brain. Not paper!

For 17-19, do not find yourself typing in terms. Learn the sum( ) and seq( ) commands on your calculator!

17. Evaluate to the nearest hundredth. [pic].

18. Evaluate to the nearest hundredth. [pic].

19. Evaluate to the nearest hundredth. How do you know when to stop adding?

[pic]

III. Integration.

Recall the method of integration by parts for products of functions: [pic]. Use the basic integration rules and any required trig identities.

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

25. [pic]

26. [pic]

27. [pic]

28. [pic]

29. [pic]

IV. Polar Coordinates. Recall that [pic], [pic], and [pic].

30. Sketch the polar curve [pic]

and determine the values of θ which generate the inner loop. You may use the POL mode on your calculator. Give these angles in exact radian measure and show work leading to your conclusion.

31. Consider the polar curves [pic]

[pic]

Locate the (x,y) coordinates of all points of intersection of the curves.

V. Application of Integration to Volumes of solids.

Your Calculus AB notes will be very helpful. Review the methods of discs, washers, shells, and cross-sections. For each problem below, graph the region, draw a picture of the solid generated, set up a definite integral to find the volume, and evaluate it exactly.

32. Let R be the region in Quadrant I bounded by the graph of y = sin x and the x-axis on the interval [0,π]. Do this without a calculator.

a) Find the area of R.

b) Revolve R about the x-axis.

c) Revolve R about the y-axis.

d) Revolve R about the line x = 2 π.

e) Let R represent the base of a solid whose cross-sections perpendicular to the

x-axis are isosceles right triangles, the hypotenuses of which lie in R.

33. Let R be the region in Quadrant I bounded by the graphs of y = e-x and y = 1 – cosx. (For x < 3. The y-axis is one boundary.) Use your calculator on this problem.

a) Find the area of R.

b) Revolve R about the x-axis.

c) Revolve R about the y-axis.

d) Revolve R about the line y = -2.

e) Let R represent the base of a solid whose cross-sections perpendicular to the

x-axis are semicircles.

f) Let R represent the base of a solid whose cross-sections perpendicular to the

y-axis are semicircles.

34. Let R be the closed region bounded by y = 5, y = 2x - 1, and the y-axis. No calculator.

a) Find the area of R.

b) Revolve R about the y-axis.

c). Find the lateral surface area of the figure in part b)

d) Find the measure of the angle, in degrees to the nearest hundredth, formed by the line y = 2x - 1 and the y-axis.

e) Find the exact volume of the solid formed if the region in part a) is revolved around the line x = 3.

VI. A few mathematical proofs.

35. Prove that[pic].

For good form, assume the truth of one side, and then transform it into the other.

Do not work to produce a statement such as 1 = 1.

Proofs by induction require two steps:

Basis step: Prove truth for n = 1.

Induction step: Assume truth for n = j, and prove truth for n = j +1.

36. Prove by induction that [pic].

37. Prove by induction that [pic] for all n > 5.

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