Summer Review Packet for Students Entering Calculus (all ...



Hello Future AP Calculus Student,

This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will be administered during the first week of classes.

You must complete the entire assignment showing sufficient evidence of effort in mathematical reasoning, use of computational skills, understanding of concepts, and communication of appropriate mathematical processes and terms. In other words, show all your work or explain how you arrived at the solution, circle your answers, label when necessary, and answer each word problem in complete sentences, etc.

It is expected that summer homework be done in a 5-subject notebook or 3-ring binder. You should only use pencil. No pens, markers, etc. allowed in notebook or binder. This will be the notebook/binder you will use through out the entire course. Calculators may be used and are recommended. You will be expected to have a TI-84 or the like but one will be offered in class if you cannot purchase one. When using calculators, show work by writing any expression that you enter into the calculator.

Remember that if this assignment is not completed to the specifications stated above, you may not be eligible to remain in the Honors Mathematics Program.

If you have any questions, feel free to email Scott.Schevling@.

All WHHS students taking an Honors or Advanced Placement (AP) course in the fall must show proof that they have successfully completed the plagiarism tutorial and correctly answered all ten questions on the plagiarism post-test at the following website: . At the end of the tutorial, you will print out the page that states that you answered all ten questions correctly and attach it to your summer assignment.

Have a great summer!

Good luck and enjoy,

The WHHS Math Department

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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