Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)
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1 Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h) m(x) Find h 6. State the domain of the following functions a. f x =!!!!!!!!!!" 3. f x = x! 2x + 3 & g x = 3x + 1 a. f (g(3)) b. (g o f )( 2) b. p x = x! + 4x + 4
2 !!! c. f x = b. y = 3x! 24x + 50!!!!"!!!" 7. Solving the following equations using completing the square. a. x! 6x 7 = 0 9. Find the inverse of each quadratic function a. y = x! + 12x + 44 b. x! + 6x = 10 b. y = 4x! 16x Convert the following to vertex form. a. y = 2x! + 4x + 5
3 10. Find the binomial factors of the following B. f x =!!!!!!!! expressions. Then solve for the roots.!!!!!!!!! a. 2x! 5x 12 = 0 b. 12x 2 + 5x 2 = 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. 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 =!!!!!!!"!!!!!!!
4 13. Test the following roots of the given polynomial using synthetic division. After testing g(x) = 2x4 + 9x 3 + 8x 2 + x +15 b. 2x + 5 the roots, completely factor the polynomial. f (x) = x 5 x 4 27x 3 + x x +120 a. x=- 1 b. x=1 c. x=- 2 d. x=- 3 e. x=3 f. x=5 g. x=- 4 c. x 4 + 6x 3 + 4x 2 6x 5 x 2 + 3x To what does the following quotient simplify? g(x) = x5 + 5x 4 15x 2 +11x 2 a. x 2 + 3x Solve the following exponential equations. ( 1 a. 8 x+1 = 32 x b. 27 )2 x+1 = (81) x+2
5 16. Use the rational zero test to identify all 20. Evaluate possible rational roots. Then test the roots using 5 3 synthetic division. Completely factor. a. 8 b f (x) = 6x x 2 + 3x Given f (x) = 4x 2 5, find f 1 (x). 17. Simplify (3x 2 y 4 z 3 ) 3 ( 2xy 3 z 2 ) Solve for x. a. 5 x = Simplify (2a 2 b 3 ) 3 (4ab 4 ) Estimate the value of the following expression. Then use the change of base formula to find the value to three decimal places. a. log 3 75 b. log 2 52 c. log Simplify ( y5 z 2 ) 3
6 24. In which quadrant will radius terminal side lie if sin(θ) < 0 and tan(θ) > 0? 28. If sinθ = 1! 2 and! cosθ = 3, find the rotation Find the reference angle of the following rotations: a. 315 b. 240 c. 150 d Express the following as functions of positive acute angles. a. sin(157 ) b. cos(222 ) c. tan(301 ) 29. What are the only two rotations in! 0 θ < 2π where! tan(θ)= Find the exact value of the following trigonometric functions. a. tan(315 ) b. cos(270 ) 30. Find the value of :! sin(π 6 )*csc(7π 6 )* tan(5π 4 )*cot(7π 4 )*cos(5π 3 )*sec(2π 3 )*sin(3π 4 )*cos(π 4 ) c. sin( 7π 6 ) d. csc(2π 3 ) e. cos(π ) f. sec( 5π 4 )
7 31. Use the rational zero test to identify all possible rational roots. Then test the roots using synthetic division. Completely factor. 33. Factor a. 4x 2 +19x +12 y = 6x 3 + x 2 31x +10 b. 6x 2 x To what does the following quotient simplify? x 4 x 3 3x 2 19x 10 x 2 3x Test the following roots of the given polynomial using synthetic division. After testing the roots, completely factor the polynomial. f (x) = x 4 2x 3 13x 2 +14x + 24 a. x=- 1 b. x=1 c. x=2 d. x=- 3 e. x=4
8 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. 37. Find m(x + h) m(x) h if m(x) = x 2 2x f (x) = 3x2 14x 5 6x 2 x Use completing the square to solve for x. a. 0 = x 2 4x Convert to vertex form of a parabola. y = x 2 + 2x 2 b. 0 = x 2 6x +13
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