Download ) = sin x sos x tan x cot x sec x csc x

Name: ___________
Date: 06-20-2016
AP Calculus Summer Assignment
Due the First Week of School (Exam. 50 pts.)
(Turn in on 09-09-2016)
Your assignment is due the first week of school and it is graded as a test. If you have trouble with any concept,
please Google, “Khan Academy”. Make sure that you show all of your work to receive full credit for your
assignment.
I. Graph each pair a,b ; c, d ; e, f ; g, h ; I, j on the same piece of graph paper. Please use radians
only in AP Calculus. Explain how they are related (i.e., what can you do to one function to obtain
the second one?). Classify these functions into odd and even function. How can you check which
trig. function is odd or even?
a.
𝑓(𝑥) = 𝑠𝑖𝑛 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
b.
𝑓(𝑥) = 𝑐𝑜𝑠𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
c.
𝑓(𝑥) = −𝑠𝑖𝑛 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
d.
𝑓(𝑥) = −𝑐𝑜𝑠𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
e.
𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
f.
𝑓(𝑥) = 𝑐𝑜𝑡 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
g.
𝑓(𝑥) = 2 𝑡𝑎𝑛 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−2𝜋, 2𝜋]
h.
𝑓(𝑥) = 2 𝑐𝑜𝑡 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
j.
𝑓(𝑥) = 𝑐𝑠𝑒𝑐 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
i.
1
𝑓(𝑥) = 𝑠𝑒𝑐 𝑥 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 [−𝜋, 2𝜋]
II. Derive the trigonometric table values for angles that are less than or equal to
special triangle, i.e.
𝜋
𝑆𝑖𝑛 ( ) =
3
√3
.
2
𝜋/6
𝜋
2
from the two
𝜋/4
?
a
2a
𝜋/3
𝜋/4
a
A
I need the exact values. Do not give me the decimal values of these trig. functions!
Trig. Function
sin x
sos x
tan x
cot x
sec x
csc x
0
𝜋/6
𝜋/4
𝜋/3
𝜋/2
Observation: Write down your observation in terms of which angles and trigonometric functions have the
same trig. values. How is this going to help you in working with trig. functions?
III. Use the above table, the reference angle, and the unit circle to find the exact value of
trigonometric function (using radian only).
2
5
a.
𝑠𝑖𝑛 (3 𝜋) = ?
b. 𝑐𝑜𝑠 (6 𝜋) =
c.
𝑐𝑜𝑠 (− 4 𝜋) = ?
3
d.
𝑠𝑒𝑐 (3 𝜋) = ?
e.
𝑐𝑠𝑐 (− 3 𝜋) = ?
8
f.
𝑐𝑜𝑠 (6 𝜋) = ?
5
9
IV. Identities and Algebraic Transformation of Expressions
Use the given identities to transform each given expression from the left side to the right side.
𝑐𝑜𝑠 2 𝑥 + 𝑠𝑖𝑛2 𝑥 = 1
cot 𝑥 =
1
tan 𝑥
=
tan 𝑥 =
sin 𝑥
cos 𝑥
=
sec 𝑥
csc 𝑥
sec 𝑥 =
1
cos 𝑥
csc 𝑥 =
1
sin 𝑥
cos 𝑥
1 + 𝑐𝑜𝑡 2 𝑥 = 𝑐𝑠𝑐 2 𝑥
1 + 𝑡𝑎𝑛2 𝑥 = 𝑠𝑒𝑐 2 𝑥
sin 𝑥
1. Prove that tan x ∙ cot x = 1.
2. Show step-by-step how you transform each expression on the left into the one on the right.
a. cos x ∙ tan x to sin x
sec x
b.
csc x ∙ tan x to
d.
csc B ∙ tan B ∙ cos B to 1
e. 𝑠𝑖𝑛2 𝛳 ∙ sec 𝛳 ∙ csc 𝛳 𝑡𝑜 tan 𝜭
f.
cot R + tan R to
g. cot D ∙ cos D + sin D to csc D
h.
csc x – sin x
i. (sec E – 1)(sec E + 1) to tan2E
j.
c. sec A ∙ cot A ∙ sin A
to 1
csc R ∙ sec R
to cot x ∙ cos x
(1 + sin B)(1 – sin B) to cos2B
k. (cos 𝝎 – sin 𝝎) to 1 – 2 cos 𝝎 sin 𝝎
2
m. (cos 𝝎 – sin 𝝎) to 1 – 2 cos 𝝎 sin 𝝎
2
o.
q.
s.
1
1−cos 𝑥
+
1
sin 𝑥 cos 𝑥
1
sin 𝑥 cos 𝑥
−
−
1
1+cos 𝑥
cos 𝑥
sin 𝑥
cos 𝑥
sin 𝑥
to 2 csc2 x
l.
n.
p.
𝑐𝑠𝑐 2 𝑥 −1
cos 𝑥
1− 𝑐𝑜𝑠2 𝑥
tan 𝑥
to
cot x ∙ csc x
to sin x ∙ cos x
(1 + sin 𝑥)(1 − sin 𝑥) to cos2x
= 𝐭𝐚𝐧 𝒙
r. 𝑠𝑒𝑐𝛳 + 𝑡𝑎𝑛𝛳 =
= 𝐭𝐚𝐧 𝒙
t. 𝑠𝑒𝑐𝛳 + 𝑡𝑎𝑛𝛳 =
𝟏
𝒔𝒆𝒄𝜭−𝐭𝐚𝐧 𝜭
𝟏
𝒔𝒆𝒄𝜭−𝐭𝐚𝐧 𝜭
x
V. Write an equation for each given graph. Show how you found both a and b values in y = a∙b .
y
Find the initial value a and the growth factor for
each graph and write the equation of the
exponential function. Label this graph as y1.
Find the initial value a and the growth factor and
write the equation of the exponential function.
Label the graph as y2.
y
Find the initial value a and the decay factor for
each graph and write the equation of the
exponential function. Label this graph as y3.
Find the initial value a and the decay factor and
write the equation of the exponential function.
Label the graph as y4.
Applications of exponential functions
1. Bacteria growth at an alarming rate. In fact every hour bacteria can split effectively doubling the number
of bacteria present. At the beginning of an experiment, a scientist has 20 bacteria in a dish. E-Coli
bacteria reproduce every 15 minutes.
(a) Write an exponential equation describing this situation.
(b) If left untreated how many bacteria cells will be present in 24 hours?
2. The population of Seattle is currently around 500,000 people. If Seattle grows by 6.5% every year,
(a) Write an exponential model for the scenario above.
(b) What will the population be in 10 years?
3. John Adam loves snowboarding. When he graduates, he would like to purchase a new Subaru so he can
go snowboarding all the time. If the Subaru that he buys costs $15,500 and depreciates at a rate of 3.6%
per year, answer the following questions about this scenario.
(a) Write an exponential equation to represent this situation.
(b) How much will the car be worth after one year?
(c) Approximately what year will Mr. Adam’s new car be
worth half as much?
VI. Work with Functions:
A. Find domain and range of each given function.
1.
𝑥
2.
𝑥 2− 9
√𝑥−2
𝑥 2 − 25
2
3. √𝑥−3
4.
𝑥+4
𝑥 3 −16𝑥
B. Find composite functions.
a. (f + g)(x)
b.
(f – g)(x)
c.
1. 𝑓(𝑥) = 𝑥 − 5 𝑎𝑛𝑑 𝑓(𝑥) = 2𝑥 3 − 4𝑥 2
2.
d
𝑓(𝑥) =
𝑥+3
𝑥−2
(a)
(b)
(c)
(d)
(e)
(f)
𝑥+3
𝑥−6
.
Is the point (2, -5/8) on the graph of f?
If x = -7, what is f(x)? What point is on the graph of f?
If f(x) = 2, what is the value of x? What points are on the graph of f?
What is the domain of f? What is the range of f?
List the x-intercepts, if any, of the graph of f?
List the y-intercept, if there is one, of the graph of f?
D. Find zeros by factoring each given function f(x).
1. 𝑓(𝑥) = −2𝑥2 + 15𝑥 − 28
2. 𝑓(𝑥) = −20𝑥2 − 110𝑥 + 400
E. Find factors for each expression.
1. 𝑧 3 − 125
2. 𝑝3 + 64
4. 3𝑦 2 − 28𝑦 − 96
5.
3.
(𝑚 + 1)3 + 8
121𝑎2 − 64
F. Simplify each complex fraction.
1.
2
−3
𝑥
2−3𝑥
2
2.
1
2
+
𝑚−1 𝑚+2
2
1
−
𝑚+2
𝑚−3
3.
𝑔
𝑎𝑛𝑑 𝑔(𝑥) =
C. Answer questions about a given function.
1. 𝑓(𝑥) =
𝑓
. ( ) (𝑥)
(f ∙ g((x))
1
1
− 2
9 𝑚
1 1
+
3 𝑚
4𝑥
3𝑥−2
G. Graph each pair of functions on the same coordinate system.
1. 𝑓(𝑥) = −√𝑥 + 3 𝑎𝑛𝑑 𝑓(𝑥) = 5 − √𝑥 + 3
2. 𝑓(𝑥) = 𝑒 𝑥 and lnx.
3. 𝑓(𝑥) = √𝑥 − 2 𝑎𝑛𝑑 𝑓(𝑥) = −2 + √𝑥 − 2
VII . Find the slope of two points and write equations of lines according to given conditions.
A. Write an equation of lines going through 2 points using point-slope form, y = y1 + m(x – x1). Convert this
form into the standard form ( ax + by = c). Some of the equation will be in the form (x = a or y = b due to
zero slope or undefined slope). Think for a second.
𝑚=
𝑦2−𝑦1
𝑥2−𝑥1
1.
A( - 4, - 2) and B(6,-2)
2.
C(4, - 7) and D(4, 8)
3.
E( - 3, - 8) and F(10, -5)
4.
G( - 2, - 1) and H(5, 0)
B. Write an equation of a line that is parallel to the given line, 2x + 4y = 12, and goes through (-2, 6)
C. Write an equation of a line that is perpendicular to the given line, 5x + 4y = - 10, and goes through (5, 0)
Have a terrific summer! Get this assignment done by 09-07=2016!
Remember to Google, “Khan Academy” if you don’t know how to do a certain
problem!