Review PAP Pre-Calculusfinal Exam 2015

Review PAP Pre-CalculusFinal Exam 2015

1. Domain & range of basic trig functions

State the domain and range for the following functions.

Domain / Range
Sin(x)
Cos(x)
Tan(x)
Cot(x)
Sec(x)
Csc (x)

2.Graphand describe the function below

a. f(x)=cos(x-)

b. f(x)=-cos(x+)

c. f(x)= 2sin(2x)

d. f(x)=2sin()

e. f(x)=-2sin(2x)

3. Name the equation of following graph

/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
a.______
/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
b.______
/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
c.______
/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
d.______
/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
e.______
/ X axis: Tick Unit is
Y axis Tick Unit is 0.5
f.______

4. Given the graph of y=f(x) below, graph and describe the following transformation

/ X axis: Tick Unit is 1
Y axis Tick Unit is 1
a.f(x)=-f(x) / b. f(x)= f(-x)
c. f(x)=2f(x+2) / d. f(x)=f(x)+2
e. f(x)=-f(2x)-2 / f. f(x)=-2f(2-x)+2

5. Find the values of the following problems if the graphs of f(x) and g(x) is given below?

/ a. f(g(4))= ?
b. f(g(2))= ?
c. g(f(1))= ?
d. g(f(-1))= ?
e. g(f(0))= ?

6. Let;;k(x)= 2x+1 ; j(x)=ln(x)+1

a.Find f◦g, g◦f, h◦f,g(h(x)), g(g(x))

b. Find the domain off◦g, h◦g, h(f(x))

c. Find the inverse of given functions above

7. Describe the following functions ( ex: domain, range , asymptotes, symmetry, even/ odd functions, one to one functions etc.)

a. f(x)= b. f(x)=c. f(x)=+5d. f(x)=

e. g(x)=g. g(x)=h. g(x)=sin(x)g. g(x)=cos(x)

Establish trig identities

I. Reciprocal Identities
/ II. Quotient Identities
/ III. Pythagorean Identities

Guidelines for Verifying Trig Identities

Begin with the most complicated side
Work on one side only
Rewrite sums or differences of quotients as one single quotient
Rewrite in terms of sine and cosine only
Factor ( GCF or Difference of Squares)
Multiply ( Foil or Distributive Property)

8. Verify the following Trig Identities

a. b.

c. d.

e. f.

g. h .

9. Exact Value

a. y= what is value of y?

b. y=what is value of y?

c. y=what is value of y?

d. y=what is value of y?

e. Given that cos θ= m, and 90 o ≤ θ 180o, what expression best represents tan θ?

f. Given that cos θ= , and 0 o ≤ θ <90o, what is exact value of tan θ?

g. Given that sin θ= , and 270 o≤ θ <360o, what is exact value of sec θ?

h. What is exact value of sec( tan-1(-1)) if 270 o ≤ θ <360o?

Law of sine and cosine

10. Find the height of the building in the figure below to the nearest foot.

11. At the end of the day, Gabe and his friends decided to go out in the dark and light some fireworks. Gabe’s grandma provided the fireworks. She told Gabe that she had been saving these bottle rockets (fireworks) ever since her childhood. Unfortunately, all the fireworks were outdated, therefore all of them were in poor condition. Tenzin, Gabe’s mom realized that all the firework devices went up in air for about 4 meters at an angle of 45◦ and descended 6.5 meters from the highest point to the ground. She proposed a question to Gabe and his friends. The question was to figure out how far it landed from the origin.

12.Triangulation can be used to find the location of an object by measuring the angles to the object from two points at the end of a baseline. Two lookouts 20 miles apart on the coast spot a ship at sea. Using the figure below find the distance, d, the ship is from shore to the nearest tenth of a mile.

13. Two lighthouses are ?? mi. apart on each side of shorelines running north and south, as shown. Each lighthouse keeper spots a boat in the distance. Lighthouse A keeper notes the location of the boat as ??˚ east of south, and the lighthouse B keeper marks the boat as ??˚ west of south. What is the distance from the boat to the lighthouse A at the time it was spotted? Round your answers to the nearest mile.

The problems of conics

14. Graph the following equations and identify the conic section

A. / / B. / / C. /

15. Find the center, circumference and area of the circle from the given endpoints of diameter

A. P (-3,7) and Q (2, -5) B. P (-3,6) and Q (3, -2)

12. Write the stand equations for with the given characteristics

Parabola

A. Focus (2,7), directrix y= -3 B. Vertex(0,0), focus (-4,0)

C.Graph the parabola below. Label the directrix, vertex, focus.

Ellipses

D. vertices (0,-12) and (0, 12); co-vertices (-5,0) and (5,0)

F.Graph the ellipse below. Label the center, vertices, co-vertices, foci.

Hyperbola

G. vertices (-7,0) and (7, 0); co-vertices (0,-8) and (0,8)

H. Graph the hyperbola below. Label the center, vertices, co-vertices, foci, and asymptotes

I. An elliptical track has a major axis that is 100 yards long and a minor axis that is 80 yards long. Find an equation for the track if its center is (0, 0) and the major axis is the y-axis.?

13.Fill in the table and sketch the parametric equation for t [-2,6]

t / -2 / -1 / 0 / 1 / 2 / 3 / 4 / 5 / 6
x
y

x =

y = 2 – t

14.For the curve + = 1 complete the following:

a. Write a pair of parametric equations for the curve.

b. Give an appropriate window and sketch the graph.

c. How could you graph just the right side of the curve?

d. How could you graph just the bottom half of the curve?

15. Eliminate the parameter to write the parametric equations as a rectangular equation.

a. x =
y = 4t + 5 / b. x = ½t + 4
y = t3 / c. . x = 3 cos t
y = 3 sin t
d. x = 4 sec t
y = 3 tan t / e. x = 4 + 2 cos t
y = -1 + 4 sin t
16. Plot the following:
A. B.
C. D.
E. F. /

17. Change the following polar equations to rectangular equations:

a. b.

c. d.

18. Change the following rectangular equations to polar equations:

a. b.

c. d.

19. Identify the polar graph (circle, spiral, cardioid, limacon, rose): If a circle, name the center (in polar coordinates) and the radius. If a limacon, name the type. If a rose, state the number of petals.

a. b. c. d.

e. f. g. f.

i. j. k.

20. Find u v . Draw a diagram to represent “tail-to-head” addition of the two vectors. Find the magnitude of the resultant vector.

a. u <-2,4 , v6,1

b. u4,0 , v1, -3

c. u <-2,-2> , v0,3

21. A boat is heading across a river at a velocity of 25 mph. The river is flowing downstream at 10 mph. What is the actual velocity of the boat?

22. An airplane is flying 340 km/hr at 12o East of North. The wind is blowing 40 km/hr at 34o South of East. What is the plane’s actual velocity?

24. Now, let’s assume that A, B, C, and D from above are each the resultant of 2 component vectors (x and y). Start by drawing each of these 4 vectors separately and then determine what 2 vectors would be needed to produce each one of them.

PAP PreCal Spring Final Exam Review 1