1! i 3$ (( )( x! 1+ i 3)

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1 Math 4C Fall 2008 Final Exam (Name) (PID) (Section) Read each question carefully; answer each question completely. Show all work: no credit for unsupported answers. Attach additional sheets if necessary. No notes allowed; use reference sheet provided. Calculator use not allowed. Each question is worth 10 points (120 total points). 1. Given x = 1+ i 3 is a root of the equation x 4! 2x 3 + x 2 + 6x! 12 = 0, find the other roots. Write f ( x) = x 4! 2x 3 + x 2 + 6x! 12 in factored form, i.e., as f ( x) = ( x! a) ( x! b) ( x! c) ( x! d). (12.R.47) The conjugate of x = 1+ i 3 (1! i 3 ) is also a root. To find the remaining roots, you may (1) synthetically divide f by 1+ i 3, and then synthetically divide this quotient by 1! i 3 to find the remaining factor ( x 2! 3), and factor this second quotient to find the last two roots ( ± 3 ), or (2) you may write factors for these two roots x! 1+ i 3 the inner parentheses x! 1! i 3 (( )( x! 1! i 3 )), and then remove (( )( x! 1+ i 3) ), and then multiply these factors ( x 2! 2x + 4 ), and divide f by this quadratic factor to find the remaining factor ( x 2! 3) and last two roots ( ± 3 ). Lastly, you need to write f in factored form as f x ( )( x! 1+ i 3) ( x! 3) ( x + 3). ( ) = x! 1! i 3 Writing conjugate root (2 points) Attempting synthetic division or writing/multiplying factors (2 points) Finding quadratic factor, x 2! 3 (2 points) Finding last two roots, ± 3 (2 points) Writing f in factored form (2 points)

2 2. Which of the following sets of data can best be represented by an exponential function, y = ab x? Find the function. [Hints: The other set of data fits a power function. Check your answer by verifying one or more values in the table for your function.] (11/3 lecture) A B x y x y Students can try fitting each set of data to an exponential function, but should eventually choose Set B. Solve for b, b = 8 = 2. To solve for a, we need to 4 find the value y( 0) which is 4. So, y = 2( 2x ) or y = 2 x+1. 2 Determining that Table B is modeled by an exponential function (3 points) Finding base b (2 points) Finding a (2 points) Finding exponential function y = 2( 2 x ) or y = 2 x+1 (3 points)

3 3. If f ( x) = x 2! 2x! 1, find (a)! f 2! 3 (3.R.11b) f 2! 3 ( ),!(b)! f ( 3! 2i),!(c)! f f ( x). ( ) = ( 2! 3) 2! 2( 2! 3)! 1 = 4! ! ! 1 = 2! 2 3 ( ) = ( 3! 2i) 2! 2( 3! 2i)! 1 = 9! 12i! 4! 6 + 4i! 1 =!2! 8i ( f ( x) ) = f x 2! 2x! 1 f 3! 2i f ( ) = ( x 2! 2x! 1) 2! 2( x 2! 2x! 1)! 1 = x 4! 4x 3 + 2x 2 + 4x + 1! 2x 2 + 4x + 2! 1 = x 4! 4x 3 + 8x + 2 Write f 2! 3 ( ) = ( 2! 3) 2! 2( 2! 3)! 1 (1 point) ( ) 2! 2( 2! 3)! 1 (2 points) Expand and simplify 2! 3 Write f ( 3! 2i) = 3! 2i Expand and simplify 2! 3 Write f ( ) 2! 2( 3! 2i)! 1 (1 point) ( ) 2! 2( 2! 3)! 1 (2 points) ( f ( x) ) = f ( x 2! 2x! 1) (1 point) ( ) = ( x 2! 2x! 1) 2! 2( x 2! 2x! 1)! 1 (1 point) ( ) 2! 2( x 2! 2x! 1)! 1 (2 points) Write f x 2! 2x! 1 Expand and simplify x 2! 2x! 1

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5 5. Find the linear function, y = mx + b, that passes through the vertices of y = x 2 + 4x and y = 2! x! 1 ( ) 2. (4.R.20) ( ) 2! 4. Find vertices of Complete the square for the first function, y = x + 2 both quadratic functions ((!2,!4),!( 1,2 )). Find slope between vertices ( m = 2 ). Either write equation in point-slope form, y = 2( x! 1) + 2, and convert to slope-intercept form, y = 2x, or write equation in slope-intercept form, y = 2x + b, and solve for b. Complete square for first function and find its vertex OR find its vertex using x =! b (3 points) 2a Find vertex of second equation (2 points) Find slope between vertices (2 points) Find equation of line in slope-intercept form (3 points)

6 ( [ ]) + D is graphed for one period in 6. A function of the form y = Asin B x! C the diagram. Determine the equation for the graph. (7.R.45) Determine vertical translation, D, by finding midpoint between maximum and minimum values, = 7. Determine period by doubling the horizontal 2 distance between the maximum and minimum points, 2( 8! 2) = 12. Determine B, B = 2! 12 or!. Determine amplitude, A, by finding difference 6 between maximum (or minimum) value and midline value, 13! 7 = 6. Determine the horizontal shift, C, graphically or algebraically. Write the equation for the graph, y = 6sin! 6 x +! or y = 6sin! 6 x 11! 6 ( + 7 or y =!6sin 6 x! 5 6 ( + 7. Determine A. (2 points) Determine B. (2 points) Determine C. (2 points) Determine D. (2 points) Write equation. (2 points)

7 7. Graph the function f ( x) = 2! x! 4. Specify the domain, range, intercepts, and any asymptotes. Determine the inverse function of f and graph it, if it exists. (5.T.1) Specify domain (! or all real numbers), range ( y >!4), and horizontal asymptote ( y =!4). Find y-intercept ( 0,!3). Solve 2! x! 4 = 0 for x-intercept (!2,0). Graph f. Find inverse function of f ( f!1 ( x) =! log 2 ( x + 4) ). Graph f!1. Domain of f. (1 point) Range of f. (1 point) Horizontal asymptote of f. (1 point) y-intercept of f. (1 point) x-intercept of f. (1 point) Graph of f. (2 points) Inverse function of f. (2 points) Graph of f!1. (1 point)

8 ( ) is a point on the line y = 2x! 1, and Q is the point (!3,3). 8. Suppose that P x, y Find P such that the length PQ is a minimum. What is this minimum distance? (4.T.12) Write equation of distance between P and Q d = x!!3 ( ( ) 2 + ( 2x + 4) ) 2 ( ( ( )) 2 + ( ( 2x! 1 )! 3 ) ) 2. Simplify radicand d = x + 3. Expand and simplify radicand ( d = 5x 2! 10x + 25 ). Complete the square for the radicand OR convert to ( ). Find P vertex form OR find vertex using x =! b d = 5 ( x! 1 ) 2 2a + 20,!V ( 1,20 ) ( 1,1). Find minimum distance 20 or 2 5. Write equation of distance between P and Q. (2 points) Expand and simplify radicand. (2 points) Complete the square for the radicand OR convert to vertex form OR find vertex using x =! b. (3 points) 2a Find P. (2 points) Find minimum distance. (1 point)

9 9. Given f ( x) = x 2! 2, find f!( 2) using the limit definition of the derivative at a f ( 2 + h) f ( 2) point, f!( 2) = lim, if it exists. (Supplemental HW 12/3) h0 Evaluate f 2 + h f ( 2)! f 2 h ( ) and ( ) f ( 2 + h) = ( 2 + h) 2! 2 = 4 + 4h + h 2! 2 = h 2 + 4h + 2 ( ( ) = 2 2! 2 = 4! 2 = 2) in numerator of limit. Simplify numerator h 2 + 4h h 2 + 4h lim = lim h!0 h h!0 h (. Simplify limit lim h!0 Evaluate limit ( lim h + 4 = 4 h!0 ). Evaluate f ( 2 + h) and f 2 Simplify numerator. (1 point) Simplify limit. (2 points) Evaluate limit. (4 points) ( ). (3 points) h 2 + 4h h = lim h + 4 h!0.

10 10. Simplify each of the following expressions. (8.T.15) a. sin!1 cos 5 6 ( ( Evaluate cos 5! 6 sin!1! 3 2 =! ( 3. b. sin( cos!1 x) cos 5! 6 = ( 3 2. Evaluate sin!1! 3 2 Draw right triangle in Quadrant I such that cos! = x = x 1. Determine opposite side ( 1! x 2 ). Write sin! sin cos!1 x Evaluate cos 5! 6. (2 points) Evaluate sin!1! 3 2. (3 points) ( ) = 1! x2 1 = 1! x 2. Draw right triangle in Quadrant I such that cos! = x = x. (2 point) 1 Determine opposite side. (1 point) Determine sine of angle. (2 points)

11 11. Prove the following identity. (8.R.103) tan! sin! cos! = 1+ tan 2! Substitute for tan! tan! = sin! cos!. Either use Pythagorean identity ( 1+ tan 2! = sec 2!) or tan 2! identity tan 2! = sin2! cos 2!. Either simplify or crossmultiply and simplify. Substitute for tan!. (2 points) Either use Pythagorean identity 1+ tan 2! = sec 2! ( ) or tan 2! identity tan 2! = sin2! cos 2!. (2 points) Simplify or cross-multiply and simplify. (6 points)

12 12. Let z 1 and z 2 denote the two square roots of!i. [DeMoivres theorem ( ), then z n = z n ( cosn! + isinn! ).] (12.R.91) states that if z = z cos! + isin! a. Compute z 1 and z 2. Write your answers in rectangular form. Write!i in trig form z = cos 3! 2 + isin 3! 2. (2 points) Use DeMoivres theorem to find the square roots of!i. (4 points) z 1 2 = cos 1 3! !k + isin 1 3! !k = cos 3! 4 + isin 3! 4 or cos 7! 4 + isin 7! 4 Convert to rectangular forms (2 points) cos 3! 4 + isin 3! 4 = ( i and cos 7! 4 + isin 7! 4 = 2 2 ( 2 2 i. b. Verify that z 1 z 2 = i. (2 points) Multiply! i 2 2! 2 2 i =! i i = i.