Math 2412 Final Exam Review

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1 Math 41 Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor and simplify the algebraic expression. 1) (x + 4) /5 - (x + 4) 1/5 1) A) (x + 4) /5 (- x - 8x - 15) B) (x + 4)((x + 4) /5 - (x + 4) 1/5 ) C) (x + 4)(- x - 8x + 15) D) (x + 4) 1/5 ((x + 4) 1/6-1) Factor the difference of two squares. ) (16x 4-1) ) A) (4x + 1)(x + 1)(x - 1) B) (4x + 1)(4x - 1) C) (4x + 1)(4x + 1) D) (x + 1) (x - 1) Factor using the formula for the sum or difference of two cubes. ) 64x + 1 ) A) (4x - 1)(16x + 4x + 1) B) (4x - 1)(16x + 1) C) (4x + 1)(16x - 4x + 1) D) prime Factor and simplify the algebraic expression. 4) (x + 9) 1/4 + (x + 9) /4 4) A) (x + 9) 1/4 (1 + (x + 9) 1/ ) B) (x + 9) 1/ (1 + (x + 9) / ) C) (x + 9) 1/ (1 + (x + 9) 1/4 ) D) (x + 9) 1/ ((x + 9) 1/ + 1) 5) (x + 4) -1/ - (x + 4) -/ 5) x + A) (x+ 4) / B) (x + 4) -1/ - (x + 4) -/ C) (x+ 4)1/ - 1 (x+ 4) 1/ D) (x+ 4)1/ - 1 (x+ 4) / Find and simplify the difference quotient f(x + h) - f(x), h 0 for the given function. h 6) f(x) = 1 5x 6) A) 0 B) -1 x (x + h) C) -1 5x (x + h) D) 1 5x 7) f(x) = x + 7x + 7) A) x + h + 7 B) 1 C) x + x + xh + h + h + 6 h D) x + h + 1

2 Evaluate the given binomial coefficient. 11 8) 8 A) 1 B) 8 C) 165 D) 990 8) Use the Binomial Theorem to expand the binomial and express the result in simplified form. 9) (x - y) 9) A) 4xy - 4xy + 4xy B) 4xy - 8xy + 4xy C) 8x - 8xy + 8xy - 8y D) 8x - 4xy + 4xy - 8y Find the term indicated in the expansion. 10) (x + y)10; 9th term 10) A) 98,415xy9 B) 95,45x8y C) 98,415x8y D) 95,45xy8 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use mathematical induction to prove that the statement is true for every positive integer n. n(n + 1) 11) n = 11) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If the given sequence is a geometric sequence, find the common ratio. 1) 4, -1, 6, -108, 4 1) A) -16 B) not a geometric sequence C) D) - Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a1, and common ratio, r. 1) Find a4 when a1 = 4, r = -. 1) A) 4 B) -108 C) -7 D) 108 Write a formula for the general term (the nth term) of the geometric sequence. 14) 1 4, - 1 0, 1 100, ,... 14) A) an = C) an = n - 1 n - 1 B) an = 1 4 n D) an = (n - 1) 5 Solve the problem. 15) A baseball player signs a contract with a starting salary of $880,000 per year and an annual increase of 4% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the sixth year? A) $1,07,845 B) $1,069,45 C) $1,070,655 D) $1,071,559 15)

3 Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located at the pole. 6 16) r = 16) 1 - cos A) hyperbola; The directrix is unit(s) to the right of the pole at x =. B) hyperbola; The directrix is unit(s) to the left of the pole at x = -. C) ellipse; The directrix is unit(s) to the left of the pole at x = -. D) ellipse; The directrix is unit(s) to the right of the pole at x =. Write the equation in terms of a rotated x'y'-system using, the angle of rotation. Write the equation involving x' and y' in standard form. 17) x + xy + y - 8x + 8y = 0; = 45 17) A) x' - 4 x'y' + y' = 0 B) x' = -4 y' C) x' = -4 y' D) x' - x'y' + y' = 0 Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form. 18) (cos 0 + i sin 0 ) 6 18) A) 1 B) -1 C) i D) -i Find projwv. 19) v = i - j; w = 5i + 1j 19) A) i j B) i j C) i j D) - 1 i j Find the angle between the given vectors. Round to the nearest tenth of a degree. 0) u = -i + 6j, v = 5i + j 0) A) B) 47.4 C) 7.4 D) 94.8 Write the complex number in rectangular form. 1) -5(cos 10 + i sin 10 ) 1) A) i B) i C) i D) i The graph of a polar equation is given. Select the polar equation for the graph. ) ) A) r = -4 cos B) r sin = - C) r = -4 sin D) r = -

4 Convert the rectangular equation to a polar equation that expresses r in terms of. ) x = 6 ) 6 A) cos = 6 B) r = C) r = 6 D) r = 6 cos sin Factor and simplify the algebraic expression. 4) (x + 7) -1/ - (x + 7) -/ 4) A) x + 6 (x+ 7) / B) (x+ 7)1/ - 1 (x+ 7) 1/ C) (x + 7) -1/ - (x + 7) -/ D) (x+ 7)1/ - 1 (x+ 7) / The point P(x, y) on the unit circle that corresponds to a real number t is given. Find the value of the indicated trigonometric function at t. 5) , 5 6 A) Find cos t. 5) B) C) 5 6 D) Use periodic properties of the trigonometric functions to find the exact value of the expression. 6) sin A) -1 B) - 1 C) - D) 6) Solve the problem. 7) The mean air temperature T, in F, at Fairbanks, Alaska, on the nth day of the year, 1 n 65, is approximated by: T = 7 sin( (n - 101)) + 5. Find the temperature at Fairbanks on day, to the 65 nearest tenth. A) -16. F B) F C) F D) F 7) Find the exact value of the expression. 8) cos -1-8) A) 5 6 B) C) 6 D) Find the exact value of the expression, if possible. Do not use a calculator. 9) sin -1 sin 4 7 9) A) 7 B) 7 C) 7 4 D) 4 7 4

5 0) cos -1 cos - 0) A) - B) C) 4 D) Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function. 1) sin(tan -1 x) 1) A) x x - 1 x - 1 B) x + 1 x + 1 C) x x + 1 D) x x + 1 x + 1 Complete the identity. ) (sin x + cos x) 1 + sin x cos x =? ) A) 1 B) 0 C) 1 - sin x D) - sec x Find the exact value of the expression. ) cos 9-18 ) A) 1 B) 1 C) D) 1 4 Use the figure to find the exact value of the trigonometric function. 4) Find cos. 4) 4 5 A) B) C) 4 5 D) 7 5 Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 5) cos 0 - sin 0 5) A) 1 B) - 1 C) - D) Use a half-angle formula to find the exact value of the expression. 6) cos 75 6) A) B) 1 - C) 1 + D) - 1-5

6 Solve the equation on the interval [0, ). 7) sin 4x = A) 4, 5 4 B) 1, 6,, 7 1, 7 6, 1 1, 5, ) C) 0,, D) 0 4 Solve the problem. 8) Two airplanes leave an airport at the same time, one going northwest (bearing 15 ) at 417 mph and the other going east at 9 mph. How far apart are the planes after 4 hours (to the nearest mile)? A) 170 miles B) 690 miles C) 760 miles D) 00 miles 8) Use Heron's formula to find the area of the triangle. Round to the nearest square unit. 9) a = 8 inches, b = 14 inches, c = 8 inches 9) A) 8 square inches B) 47 square inches C) 49 square inches D) 0 square inches Convert the rectangular equation to a polar equation that expresses r in terms of. 40) y = x 40) A) r (cos + sin ) = B) r = cot x cscx C) r = 9 cot x cscx D) r = cot x Find the indicated sum. 41) 4 k = 1 (-1)k(k + 11) 41) A) 46 B) 54 C) -54 D) Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 4) ) A) 6 i B) 6 i C) 6 i D) 6 i i = 1 i = 1 i = 1 i = 0 Find the indicated sum. 4) Find the sum of the first 70 terms of the arithmetic sequence: 19, 14, 9, 4,... 4) A) -1 B) -10,745 C) -10,740 D) -10,90 Find the term indicated in the expansion. 44) (4x + y)10; 7th term 44) A) 9,191,040x4y6 B) 1,06,680x6y4 C) 9,797,760x6y4 D) 1,06,680x4y7 6

7 Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. 45) lim x 1 x + 7x - x - 4x + 45) A) does not exist B) C) 8 D) 0 Find the derivative of f at x. That is, find f '(x). 46) f(x) = x - 8x + 11; x = 46) A) -8 B) 4 C) -4 D) -6 The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. 47) a. lim x 0 f(x) b. f(0) 47) A) a. lim x 0 f(x) = 1 b. f(0) does not exist C) a. lim f(x) = 0 x 0 b. f(0) = 0 B) a. lim x 0 f(x) does not exist b. f(0) does not exist D) a. lim f(x) = -1 x 0 b. f(0) does not exist Find the standard form of the equation of the ellipse satisfying the given conditions. 48) Major axis horizontal with length 1; length of minor axis = 6; center (0, 0) 48) A) x 6 + y 9 = 1 B) x y x = 1 C) y x = 1 D) y 9 = 1 Find the location of the center, vertices, and foci for the hyperbola described by the equation. 49) (x + ) - 100(y - ) = ) A) Center: (-, ); Vertices: (10, ) and (-10, ); Foci: (- 101, ) and ( 101, ) B) Center: (-, ); Vertices: (-1, ) and (8, ); Foci: ( , ) and ( , ) C) Center: (, -); Vertices: (-7, -) and (1, -); Foci: ( - 101, ) and ( + 101, ) D) Center: (-, ); Vertices: (-1, ) and (7, ); Foci: ( , ) and ( , ) 7

8 Solve the problem. 50) A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is feet. If the distance across the top of the mirror is 80 inches, how deep is the mirror in the center? A) in. B) in. C) in. D) in. 50) Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations. 51) x = t + 1, y = t - 1; - t 51) A) y = x - ; -7 x 9 B) y = -x ; -4 x 4 C) y = -x - ; -7 x 9 D) y = x ; - x 1 Solve the problem. Round to the nearest dollar if needed. 5) To save for retirement, you decide to deposit $500 into an IRA at the end of each year for the next 40 years. If the interest rate is 8% per year compounded annually, find the value of the IRA after 40 years. A) $51,811 B) $15,10,10 C) $647,641 D) $597,5 5) Find the indicated sum. 5) Find the sum of the first 0 terms of the arithmetic sequence: 14, 7, 0, -7,... 5) A) -16 B) -110 C) D) Find the quotient z 1 of the complex numbers. Leave answer in polar form. z 54) z1 = 0(cos 40 + i sin 40 ) z = 5(cos 7 + i sin 7 ) 54) A) 6(cos + i sin ) B) 6(cos 47 + i sin 47 ) C) 5(cos - i sin ) D) 5 cos i sin 7 7 Solve the problem. 55) Two airplanes leave an airport at the same time, one going northwest at 404 mph and the other going east at 47 mph. How far apart are the planes after hours (to the nearest mile)? A) 694 miles B) 1659 miles C) 1774 miles D) 08 miles 55) 8

9 Answer Key Testname: MATH41_FALL016_FINAL_EXAM_REVIEW 1) A ) A ) C 4) A 5) D 6) C 7) A 8) C 9) D 10) D 11) S1: 1 1) D 1) B 14) C 15) C 16) B 17) B 18) B 19) B 0) D 1) B ) C ) B 4) D 5) D? =? = = ( 1)(1 + 1) Sk: k = k(k + 1) Sk+1: (k + 1) = (k + 1)(k + ) We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely (k+1), to both sides." k + (k + 1) = (k + 1) = (k + 1) = (k + 1) = k(k + 1) k(k + 1) + (k + 1)(k + 4) (k + 1)(k + ) + (k + 1) 4(k + 1) We have shown that if we assume that Sk is true, and we add ((k+1) to both sides of Sk, then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n. 9

10 Answer Key Testname: MATH41_FALL016_FINAL_EXAM_REVIEW 6) C 7) C 8) A 9) B 0) B 1) D ) A ) C 4) A 5) A 6) B 7) B 8) C 9) D 40) B 41) D 4) B 4) B 44) A 45) C 46) C 47) A 48) A 49) D 50) C 51) A 5) C 5) D 54) A 55) D 10