Winter 2014 Practice Final 3/21/14 Student ID

Transcription

1 Math 4C Winter 2014 Practice Final 3/21/14 Name (Print): Student ID This exam contains 5 pages (including this cover page) and 20 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You may not use your books, notes, or any calculator on this exam. You are required to show your work on each problem on this exam. The following rules apply: Organize your work, in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit. Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. If you need more space, use the back of the pages; clearly indicate when you have done this. This practice exam is much much longer than the real final exam. It is provided to make you more familiar with the types of questions that may appear on the real midterm exam.

2 Math 4C Practice Final - Page 2 of 5 3/21/14 1. Do all the questions on Practice Exam 1, Exam 1, Practice Exam 2, and Exam (a) Find the area of a triangle containing an angle of π/3 with adjacent sides of lengths 4 and 5. (b) Suppose that a parallelogram has area 75 and sides of lengths 10 and 15. What are the measures of the angles in this parallelogram? 3. Suppose that a regular decagon is inscribed in a circle with radius r. What is the area of the decagon? 4. (a) Find sin( π/8) exactly using a half-angle formula. (b) Given that sin(18 ) = ( 5 1)/4, what is sin(36 )? 5. (a) Give the period and amplitude of f(x) = 3 sin(5x 1) + 2 and g(x) = cos(2x/3)/5. (b) Suppose that f(x) is a periodic function whose maximum is 150 and minimum is 80. What is its amplitude? 6. (a) How far has the graph of y = 3 cos(2x) been shifted to the right to obtain the graph y = 3 cos(2x π/8)? What fraction of its period has it been shifted? (b) How far has the graph of y = (1/2) sin(x/3) been shifted to the right to obtain the graph y = (1/2) sin(x/3 π/10)? What fraction of its period has it been shifted? 7. Write the following points in polar coordinates: ( 1, 0), (3, 3), ( 4 3, 4), (0, 5). 8. The following points are written in polar coordinates. Write them in rectangular coordinates: (8, π/2), (2, 5π/4), (1/2, π/6), (0, 0). 9. Give the center and radius of the circle r = 8 cos(θ). 10. Write the following complex numbers in the form a + bi. (a) (8 + i) (4 4i) (b) (3 + i)(2 5i) (c) 8 7i (d) 1/(2 + 2i) (e) (3 + 2i)/(1 i). 11. (a) Are the following true or false. 1. Every polynomial with real coefficients has a complex root. 2. If z is a complex root of a polynomial f(x) with real coefficients, then so is z. (b) Find the (complex) roots of f(x) = x 2 + x Write the following complex numbers in polar coordinates: 3, 2i, 4 + 4i, 3 3i. 13. If z = r(cos(θ) + i sin(θ)), what is z? 14. (a) Suppose that z = r(cos(φ) + i sin(φ)) and w = s(cos(θ) + i sin(θ)). Also suppose that φ + θ is an integer multiple of 2π. What is zw? (Hint: use Euler s Formula.) (b) Suppose that z = cos(π/6) + i sin(π/6). What is the smallest positive integer n such that z n = 1? (Hint: use Euler s Formula.)

3 Math 4C Practice Final - Page 3 of 5 3/21/ (a) What is the 10th term in the arithmetic sequence which starts 4, 7,...? (b) What is the n-th term in the arithmetic sequence whose second term is 6 and whose fifth term is 33? (c) What is the 10th term in the geometric sequence which starts 4, 20,...? (d) What is the n-th term in the geometric sequence whose second term is 6 and whose fifth term is 33? 16. (a) The Fibonacci sequence is defined by Write the first 8 terms of this sequence. a 1 = 1, a 2 = 1, a n+2 = a n+1 + a n for n 1. (b) Consider the arithmetic sequence starting at 5 with constant step-size 4. Write this sequence recursively. 17. Compute the following sums: (a) (b) (c) (d) (e) 4 k(k 1) (7 + 3k) k (a) Compute the following binomial coefficients. 1. ( ) ( ) 10 7

4 Math 4C Practice Final - Page 4 of 5 3/21/14 3. ( n ) n 1 (b) What is the 6th row of Pascal s triangle? (c) What is (x + y) 7? 19. Give the following limits. (a) lim 5/n n (b) lim n (.99)n (c) lim n ( 1)n (d) lim n 1.01n (e) lim n (n3 n 1)/(3n 3 + n 2 5) 20. Compute the following infinite series. (a) 1/3 + 1/9 + 1/27 +. (b) 3 (1/2) n. n=1

5 Math 4C Practice Final - Page 5 of 5 3/21/14 Formula sheet 1. Base change in logarithms:. log b y = log a y log a b 2. Compound interest, compounded n times per year: M(t) = P ( 1 + r ) nt n 3. Compound interest, compounded continuously: M(t) = P e rt 4. Double angle formulas: (a) cos(2θ) = 1 2 sin 2 (θ) = 2 cos 2 (θ) 1 = cos 2 (θ) sin 2 (θ) (b) sin(2θ) = 2 cos(θ) sin(θ) (c) tan(2θ) = 2 tan(θ)/(1 tan 2 (θ)) 5. Half angle formulas: (a) cos(θ/2) = ± (1 + cos(θ))/2 (b) sin(θ/2) = ± (1 cos(θ))/2 (c) tan(θ/2) = (1 cos(θ))/ sin(θ) = sin(θ)/(1 + cos(θ)) 6. Euler s formula: 7. Sum of an arithmetic series: 8. Sum of a geometric series: e iθ = cos(θ) + i sin(θ) (F + L) N 2 b 1 rn 1 r