AP CALCULUS SUMMER PACKET

AP CALCULUS SUMMER PACKET Name: For students to successfully complete the objectives of the course, the students must demonstrate a high level of inde...
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AP CALCULUS SUMMER PACKET Name: For students to successfully complete the objectives of the course, the students must demonstrate a high level of independence, capability, dedication and effort. This packet will help you improve your skills. Complete this packet on your own and submit it on the first day of the school year. Guidelines: • All work must be shown in the packet or in a separate paper attached to the packet • Be sure all f)i-oblems are neatly organized and all writing is legible • I expect you to come in with certain understandings that ar -l-prerequisite to Calculus. A list of this topical understandings are listed below. complex fraction systems of equations inverse function trigonometric function exponents graphing

function intercepts interval notation domain and range equation of a line radian and angle measure inverse trig function circle and ellipse logarithm factoring

Formula Sheet Reciprocal Identities:

cscx

Quotient Identities:

tan x =

Pythagorean Identities:

sin2 x + cos2 x = I

Double Angle Identities:

sin 2x = 2 sin x cos x 2 tan x tan 2x = 1 – tan2 x

Logarithms: y = loge x is equivalent to

sin x sin x cos x

sec x =

1 cos x

cot x =

cos x sin x

cot x =

tan 2 x + I = sec2 x

tan x

I + cot2 x = csc2 x

cos 2x = cos2 x – sin2 x = I –2sin2 x = 2 cos2 x – I The Zero Exponent: x°=1, for x not equal to 0.

x = aY Multiplying Powers

Product property:

log,, mn = log, m + logb n

Quotient property:

m log,, — = log, in – log, n

Multiplying Two Powers of the Same Base:

(xa)(xb)

x(a+b)

Multiplying Powers of Different Bases:

(Xy)a = (Xa)(ya)

Power property:

log,, m'' = p log,, in

Property of equality: then m n

If log, in = log,, n,

Dividing Powers Dividing Two Powers of the Same Base:

( ia)/(xb) =

Dividing Powers of Different Bases:

(X/y)a = (xa)/(ya)

Change of base formula:

• log, n loge n = logb a

Fractional exponent:

=

Negative Exponents:

=)1Ixii

Slope-intercept form: y = mx + b

2

Point-slope form:

y = m(x – x, )+

Standard form:

Ax + By + C --• 0

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:

—7 6 x +1 5 x +1

7

6 x +1. x +1 5 • x +1 x +1

—7x-7-6 5

—7x-13 5

—2 3x —2 3x + + x x-4 = x x-4 . x(x— 4) = —2(x— 4) + 3x(x) 1 1 x(x— 4) 5(x)(x— 4)-1(x) 5 5 x4 x- 4

-2x + 8+ 3x2 5x2 —20x— x

=

3x2 — 2x+ 8 5x2 —21x

Simplify each of the following.

25 --a 1.

a

5±a

4 x+ 2 2. 10 5+ x+ 2

12 2x-3 3. 15 5+ 2x —3

2-

4

2x 3x-4 5. 32 x+ 3x— 4 1—

3

Function To evaluate a function for a given value, simply plug the value into the function for x. Recall: (f g)(x) = f (g(x)) OR f[g(x)] 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 f (x) =2 x2 +1 and g(x) = x — 4 find f(g(x)). f(g(x))= f (x —4)

= 2(x — 4)2 + 1 = 2(x2 — 8x +16) + 1 =2x2 -16x+32+1 f (g(x)) = 2x2 —16x +33

Let f (x) = 2x +1 and g(x) = 2x2 —1. Find each. 8. f (t +1) =

6. f (2) =

7. g(-3) =

9. f[g(-2)]=

10. g[f (m + 2)].

Let 12. f

11

f

f (x)= sinx Find each exactly. Tr

)=

13.

3

Let f (x) = x2, g(x) = 2x + 5, and h(x) = x2 —1. Find each. 14. h[f(-2)]=

15. f[g(x —1)]=

4

16. g[h(x3)]=

f (x)

Systems

Use substitution or elimination method to solve the system of equations. Example: x2 +y-16x+39= 0 X2 -

y2 — 9 = 0

Elimination Method

Substitution Method

2x2 —16x + 30 = 0

Solve one equation for one variable.

x2 -8x+15 = 0 (x— 3)(x — 5) = 0 x=3andx=5 Plug x=3 and x= 5 into one original 52_ y2 _ 9 0 32 _ y2 9_ 0 —y2 = 0 16= y2 y = ±4 y=0 Points of Intersection (5,4), (5,-4-) and (3,0)

y2 = —x2 +16x — 39

(1st equation solved for y)

x2 — (—x2 +16x — 39)— 9 = 0 Plug what y2 is equal to into second equation. 2x2 —16x+ 30 = x 2 8x+15 = 0

(The rest is the same as previous example)

(x —3)(x— 5) = 0 x= 3 or x-5

Find the point(s) of intersection of the graphs for the given equations.

23.

x+y= 8 4x — y =7

24.

x2 + y =6 x+y=4

25.

x2 — 4y2 — 20x — 64y —172 = 0 16x2 +4y2 — 320x + 64y +1600 = 0

Interval Notation

26. Complete the table with the appropriate notation or graph. Solution

—2 < x

Interval Notation

Graph

4 [-1,7)

--/8

Also, recall that to PROVE one function is an inverse of another function, you need to show that: f (g(x)) = g(f (x)) = x Example: x–9 If: f (x) = 4 and g (x) = 4x + 9 show f(x) and g(x) are inverses of each other.

g(f (x)) = 4

(x-0 +9 4

=x-9+9 =

f (g(x)) =

(4x + 9) – 9

4 4x +9 –9 4 = 4x 4 -= x f (g(x)) = g(f (x)) = x therefore they are inverses of each other.

Provef and g are inverses of each other. x3 36. f (x) = — 2

g(x) = VZX

37. f (x) = 9 – x2 , x

0

g(x) =

x

Radian and Degree Measure 180° to get rid of radians and gradians convert to degrees.

gradians to get rid of degrees and 180° convert to radians.

Use

57c

Use

L

47r

46. Convert to degrees:

a. — 6

▪.

47. Convert to radians:

a. 45'

b. —17

c. 2.63 radians

5

c. 237

Angles in Standard Position 48. Sketch the angle in standard position. 5TC C. --

b. 230°

3

10

d. 1.8 radians

Reference Triangles 49. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. 2 a. — 7C 3

b. 225

7C

d. 30

C. --

4

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:- sin 900 =1

50.

cos— = u 2

a.) sin180°

b.) cos 270°

c.) sin(-90°)

d.) sing'

e.) cos 360°

f.) cos(—r) 11

Graphing Trig Functions

y = sin x and y = cos x have a period of 2 r and an amplitude of I. Use the parent graphs above to help you — = period, sketch a graph of the functions below. For f (x). A sin(Bx + C)+ K, A = amplitude, 27c

c — = phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift. Graph two complete periods of the function. 51.

f(x) = 5 sinx

53. f (x) = cosi

52. f (x) = sin 2x

ir 4

54. f (x) = cos x — 3

Trigonometric Eq_uations: Solve each of the equations for 0 5_ x < 2r. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 5_ x < 22r. 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 beginning of the packet.) 1

55. sinx = -2

56. 2cosx = Nrj.

12

57. cos 2x =

1 N/2

58. sin2 x =

2

59. sin 2x = -2

60. 2 cos2 x —1— cos x = 0

61. 4cos2 x — 3 = 0

62. sin2 x + cos 2x — cos x = 0

Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsin (x)

sin-1 (x)

irivers.e trig fimations are defined only in the quadrants as indicated below clue to their restricted dornainS. cos-lx 0 cos-1 x >0 tan-1 x >0

41 sin-1 x