NAME ___________________________________ YEAR and SECTION ______________________
Calculus’First’Semester’Review’’’’’’’’’’Name...
Transcript of Calculus’First’Semester’Review’’’’’’’’’’Name...
Calculus First Semester Review Name: Section: Evaluate the function: 1. 𝑓 𝑥 = 4𝑥! − 2 a. 𝑓(−2) b. 𝑓(𝑥 − 1)
c. 2. 𝑚 𝑥 = 3𝑥! + 2𝑥 − 1
Find 3. 𝑓 𝑥 = 𝑥! − 2𝑥 + 3 & 𝑔 𝑥 = 3𝑥 + 1 a. b.
c. d. 4. 𝑔 𝑥 = 3𝑥! + 1 Find 𝑔!!(𝑥) 5. 𝑝 𝑥 = 4𝑥! + 2 Find 𝑝!!(𝑥) 6. State the domain of the following functions a. 𝑓 𝑥 = !!!
!!!!!!!"
b. 𝑝 𝑥 = 𝑥! + 4𝑥 + 4
f (x + h)− f (x)h
m(x + h)−m(x)h
f (g(3)) (g o f )(−2)
(g o f )(x + 2) ( f ( f (x))
Calculus First Semester Review Name: Section: c. 𝑓 𝑥 = !!!
!!!!"!!!"
7. Solving the following equations using completing the square. a. 𝑥! − 6𝑥 − 7 = 0 b. 𝑥! + 6𝑥 = −10 8. Convert the following to vertex form. a. 𝑦 = 2𝑥! + 4𝑥 + 5
b. 𝑦 = 3𝑥! − 24𝑥 + 50 9. Find the inverse of each quadratic function a. 𝑦 = 𝑥! + 12𝑥 + 44 b. 𝑦 = 4𝑥! − 16𝑥 + 5
Calculus First Semester Review Name: Section: 10. Find the binomial factors of the following expressions. Then solve for the roots. a. 2𝑥! − 5𝑥 − 12 = 0 b. 12x2 + 5x − 2 = 0 11. a. Find the domain of the following function. b. Identify if the limited domain values are vertical asymptotes or holes. c. If it is hole, identify the location of the hole. If it is a vertical asymptote, write the equation of the asymptote. d. Write the equation of the horizontal asymptote. A. 𝑓 𝑥 = !!!!!!!"
!!!!!!!
B. 𝑓 𝑥 = !!!!!!!!!!!!!!!!!
12. Sketch the graph of 𝑔 𝑥 = !!!!!!
!!!!.
Find: Vertical asymptotes; horizontal asymptotes; holes; and the x-‐ and y-‐intercepts. Clearly show all algebraic work and label these values on the graph.
Calculus First Semester Review Name: Section: 13. Test the following roots of the given polynomial using synthetic division. After testing the roots, completely factor the polynomial.
a. x=-‐1 b. x=1 c. x=-‐2 d. x=-‐3 e. x=3 f. x=5 g. x=-‐4 14. To what does the following quotient simplify?
a.
b.
c. 15. Solve the following exponential equations.
a. b.
f (x) = x5 − x4 − 27x3 + x2 +146x+120
g(x) = x5 + 5x4 −15x2 +11x − 2
x2 + 3x −1
g(x) = 2x4 + 9x3 + 8x2 + x+15
2x+ 5
x4 + 6x3 + 4x2 − 6x − 5x2 + 3x+ 2
8x+1 = 32x( 127)2x+1 = (81)x+2
Calculus First Semester Review Name: Section: 16. Use the rational zero test to identify all possible rational roots. Then test the roots using synthetic division. Completely factor.
17. Simplify
18. Simplify
19. Simplify
20. Evaluate
a. b.
21. Given , find . 22. Solve for x. a. 23. Estimate the value of the following expression. Then use the change of base formula to find the value to three decimal places. a. b. c.
f (x) = 6x3 + 29x2 + 3x −10
(3x2y4z3 )3(−2xy3z2 )4
(2a2b3 )3
(4ab4 )5
( y5
z2)−3
853 81
−34
f (x) = 4x2 − 53 f −1(x)
5x = 300
log3 75 log2 52 log5 200
Calculus First Semester Review Name: Section: 24. In which quadrant will radius terminal side lie if sin(θ ) < 0 and tan(θ ) > 0 ? 25. Find the reference angle of the following rotations: a. 315° b. 240° c. 150° d. 32° 26. Express the following as functions of positive acute angles. a. sin(157°) b. cos(222°) c. tan(301°) 27. Find the exact value of the following trigonometric functions. a. tan(315°) b. cos(270°)
c. sin(7π6) d. csc(2π
3)
e. cos(π ) f. sec(5π4)
28. If !sinθ = 12 and !
cosθ = − 32 , find the rotation.
29. What are the only two rotations in !0≤θ <2π where !tan(θ )= − 3 . 30. Find the value of :
!sin(π6 )*csc(
7π6 )* tan(
5π4 )*cot(
7π4 )*cos(
5π3 )*sec(
2π3 )*sin(
3π4 )*cos(
π4 )
Calculus First Semester Review Name: Section: 31. Use the rational zero test to identify all possible rational roots. Then test the roots using synthetic division. Completely factor.
32. To what does the following quotient simplify?
33. Factor a. b. 34. Test the following roots of the given polynomial using synthetic division. After testing the roots, completely factor the polynomial.
a. x=-‐1 b. x=1 c. x=2 d. x=-‐3 e. x=4
y = 6x3 + x2 − 31x+10
x4 − x3 − 3x2 −19x −10x2 − 3x − 2
4x2 +19x+12
6x2 − x − 2
f (x) = x4 − 2x3 −13x2 +14x+ 24
Calculus First Semester Review Name: Section: 35. a. Find the domain of the following function. b. Identify if the limited domain values are vertical asymptotes or holes. c. If it is hole, identify the location of the hole. If it is a vertical asymptote, write the equation of the asymptote. d. Write the equation of the horizontal asymptote.
36. Use completing the square to solve for x. a. b.
37. Find if 38. Convert to vertex form of a parabola.
f (x) = 3x2 −14x − 56x2 − x −1
0 = x2 − 4x+1
0 = x2 − 6x+13
m(x + h)−m(x)h m(x) = x2 − 2x
y = x2 + 2x − 2