MAC 2312 Midterm: Test 2 study guide

1. Let : 1 +2i, w 2-i, and 4+3i. Compute (a) : + 3w;(b) -2w+ Use the quadratic formula to solve these equations; express the answers as (d) zz-z = 1; (e) z-22. 3. Sketch the locus of those points w with (a)|w 3:(b)lw-21- 1:(e)|w +22 4. Find Re(1/2) and Im(1/z) if z = x + iy, 0. Show that Re(iz)--Im z and 5. Give the polar representation for (a)-1+ i; (b) I + iV3; (c)-|; (d) (2-i) 6. Give the complex number whose polar coordinates (r, 0) are (a) (V3, x/4); 7, Let a, b, and c be real numbers with a ψ 0 and b2 4ac. Show that the two roots 8. Suppose that A is a real number and B is a complex number. Show that (g) Re[zl1 +] Im(iz)- Re z. (b) (1 y 2, π); (c) (4,-π/2); (d) (2,-24); (e) (1,4ç §(f) (v/2, 9-4). of ax + bx c0 are complex conjugates of each other. and 9. Show that |z 1 if and only if 1/z- 10. Let z and w be complex numbers with zw 0. Show that either z or w is zero. 11. Show that lz + w12-12-w12 = 4 Re(zi) for any complex numbers z, w. 12. Let z1, z2 z, be complex numbers. Establish the following formulas by mathematical induction: (b) Re(zit zit , , , + z.) = Re(z.) + Reiz) + + Re(%) 13. Determine which of the following sets of three points constitute the vertices of 14. Show that cos θ-cos ψ and sine-sin ψ if and only if θ-ψ is an integer 15. Show that the triangle with vertices at 0, z, and w is equilateral if and only if a right triangle: (a) 3+5i, 2+2i,5 +i; (b)2+ i, 35i,4 + i; (c) 6+4i,7+Si, 8 + 4i. multiple of 2 10 Chapter 1 The Complex Plane 16. Let zo be a nonzero complex number. Show that the locus of points tzo. -ao

nental identities to simplify the expressi tan θ cot θ A) tan20 Objective 2) Simplity Trigonometric Espressin I B) cos3e C) sec2e D) sin θ Perform the indicated 2)sec 0-sec θ operations and simplify the result so there are no quotients. A)sin θ tan θ btive () Perform operations with trigonometric expressions and simplify the results. B) -2 tan20 C)1+cot θ se an appropriate identity to find the exact value of the expression. 3) sin (165) A)266-1 B) 203+1 3-1 C) D) Okyes tive、(73) Find Exact Value of Expression (Degrees) se the cofunction identities to find an angle 0 that makes the statement true. 4)sin (30-17°) = cos (θ + 43°) A) θ-16" Obpective. (7.3) Use Cofunction Identities to Find Angle B)θ=90° C0-10 D) θ = 60 e an identity to write the expression as a single trigonometric function or as a single number 5) 2 cos2 22.5-1 A) V3 1 2 C) 业 D) Objective: (1) Use double-angle identities. identities to find the indicated value for each angle measure. 21 29 6)sin θ cos θ >0 Find cos(20). 41 841 A) B) C) D) 841 841 841 Objective: (1) Use double-angle identities. the product as a sum or difference of trigonometric functions. 7) cos 38° sin 23° A) cos 61°-cos 15°) B) cos 61°+cos 15) C) sin 61° + sin 15°) D) (sin 61-sin 15") Obiective (2) Use product-to-sum and sum-to-product identities

Need Help?ReWalch Talk to a Tulb /1 points sCalcET8 3.4.007 Find the derivative of the function Fx) (3x6 + 4x3)4 F(x) = Need Help? Resd it Watch It Talk to a Tutor +1 points SCalcETS 3 4.009 Find the derivative of the function. f(x) = v 9x + 8 f (x) Read It Watch It Talk to a Tutor Type here to search

34.0 Find the derivative of the function /(x) = (2x-6)4(x2 + x + 1)5 f(x) Need Help? Read t

0/1 points I Previous Answers SCalcET8 3.4.031 Find the derivative of the function.E F(t) = e3tsin(2t) F(t) = | e 3tsin 2t (6t cos2+ 3 sin 2t ) | Need Help? L-Read it | | |L-Watchltー」11 Talk to a Tutor 9、0-11 points scalcET8 3 4 053 Find an equation of the tangent line to the curve at the given point. y = sin(sin(x)), (r, 0) Need Help? aditWatch t Talk to a Tutor

-12 points SCalcET8 3.4.063 A table of values for f, g, f, and g' is given. xf(x) g(x) f'(x) g'(x) 2 4 6 7 9 2 3 2 (a) If h(x)=rg(x), find h'(3). h'(3) (b) If H(x) g(f(x), find H'(2). H(2) Need Help? Read It WatehTak to a Tutor Talk to a Tutor Submit Assignme

Write the composite function in the form Ag(x)). [Identify the inner function u = g(x) and the outer function y = ru).] 3 (g(x), f(u)) = ( | 277