Summer Review Packet for Students Entering Calculus (all ...
Walt Whitman HIGH SCHOOL
SUMMER REVIEW PACKET
For students in entering AP CALCULUS BC
Name: __________________________________________________________
1. This packet is to be handed in to your Calculus teacher on the first day of the school year.
2. All work must be shown in the packet OR on separate paper attached to the packet.
3. Completion of this packet is worth homework points and will be counted in your first marking period grade.
Summer Review Packet for Students Entering Calculus
Complex Fractions
When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction.
Example:
[pic]
[pic]
Simplify each of the following.
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic]
Functions
To evaluate a function for a given value, simply plug the value into the function for x.
Recall: [pic] read “f of g of x” Means to plug the inside function (in this case g(x) ) in for x in the outside function (in this case, f(x)).
Example: Given [pic] find f(g(x)).
[pic]
Let [pic]. Find each.
6. [pic] ____________ 7. [pic]_____________ 8. [pic] __________
9. [pic]__________ 10. [pic]___________ 11. [pic]______
Let [pic] Find each exactly.
12. [pic]___________ 13. [pic]______________
Let [pic]. Find each.
14. [pic] 15. [pic] 16. [pic]
Find [pic] for the given function f.
17. [pic] 18. [pic]
Intercepts and Points of Intersection
To find the x-intercepts, let y = 0 in your equation and solve.
To find the y-intercepts, let x = 0 in your equation and solve.
Example: [pic]
[pic] [pic]
Find the x and y intercepts for each.
19. [pic] 20. [pic]
21. [pic] 22. [pic]
Use substitution or elimination method to solve the system of equations.
Example:
[pic]
Find the point(s) of intersection of the graphs for the given equations.
23. [pic] 24. [pic] 25. [pic]
Interval Notation
26. Complete the table with the appropriate notation or graph.
Solution Interval Notation Graph
[pic]
[pic]
8
Solve each equation. State your answer in BOTH interval notation and graphically.
27. [pic] 28. [pic] 29. [pic]
Domain and Range
Find the domain and range of each function. Write your answer in INTERVAL notation.
30. [pic] 31. [pic] 32. [pic] 33. [pic]
Inverses
To find the inverse of a function, simply switch the x and the y and solve for the new “y” value.
Example:
[pic]
Find the inverse for each function.
34. [pic] 35. [pic]
Also, recall that to PROVE one function is an inverse of another function, you need to show that:
[pic]
Example:
If: [pic] show f(x) and g(x) are inverses of each other.
[pic]
Prove f and g are inverses of each other.
36. [pic] 37. [pic]
Equation of a line
Slope intercept form: [pic] Vertical line: x = c (slope is undefined)
Point-slope form: [pic] Horizontal line: y = c (slope is 0)
38. Use slope-intercept form to find the equation of the line having a slope of 3 and a y-intercept of 5.
39. Determine the equation of a line passing through the point (5, -3) with an undefined slope.
40. Determine the equation of a line passing through the point (-4, 2) with a slope of 0.
41. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of 2/3.
42. Find the equation of a line passing through the point (2, 8) and parallel to the line [pic].
43. Find the equation of a line perpendicular to the y- axis passing through the point (4, 7).
44. Find the equation of a line passing through the points (-3, 6) and (1, 2).
45. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3).
Radian and Degree Measure
Use [pic] to get rid of radians and Use [pic] to get rid of degrees and
convert to degrees. convert to radians.
46. Convert to degrees: a. [pic] b. [pic] c. 2.63 radians
47. Convert to radians: a. [pic] b. [pic] c. 237[pic]
Angles in Standard Position
48. Sketch the angle in standard position.
a. [pic] b. [pic] c. [pic] d. 1.8 radians
Reference Triangles
49. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible.
a. [pic] b. 225[pic]
c. [pic] d. 30[pic]
Unit Circle
You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate of the circle is the cosine and the y-coordinate is the sine of the angle.
Example: [pic]
50. [pic] [pic]
[pic] [pic]
[pic] [pic]
Graphing Trig Functions
y = sin x and y = cos x have a period of 2[pic] and an amplitude of 1. Use the parent graphs above to help you sketch a graph of the functions below. For [pic], A = amplitude, [pic]= period,
[pic]= phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift.
Graph two complete periods of the function.
51. [pic] 52. [pic]
53. [pic] 54. [pic]
Trigonometric Equations:
Solve each of the equations for [pic]. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, [pic]. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at the end of the packet.)
55. [pic] 56. [pic]
57. [pic] 58. [pic]
59. [pic] 60. [pic]
61. [pic] 62. [pic]
Inverse Trigonometric Functions:
Recall: Inverse Trig Functions can be written in one of ways:
[pic]
Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains.
cos-1x < 0 sin-1x >0
cos-1 x >0
tan-1 x >0
sin-1 x ................
................
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