MATH 1172 Lecture Notes - Lecture 12: Thai Baht

1. Find the exact area of the region enclosed by one loop of the 3-leaved rose r-4 cos(3 e) 2 Find the slope of the tangent line to the given polar curve at the point specified by the value of e. r-cos( 30)·0-4 a 1 b.2 e.3 d4 e.None of these 3 Consider the parametric equationsx-3ry-3. Find the point on the curve where the tangent i horizontal a. (0,3) b (0,-3) c (-1.-3) d. (1,0) e. None of these Find the exact length of the polar curve r-3 cos(0),0 θ π a.π b, 2 π c, 3 π d, 4 π e" None ofthese 5 Determine whether the sequence {%) converges or diverges. If it converges, find the limit. a. It is divergent b. It is convergent, the limit ise c. It is convergent, the limit is 2 d. It is convergent, th is I e. None of these Determine whether the series Σ( 6, is convergent or divergent. If it is convergent, sum. a It is divergent b. It is convergent, the sum i c. It is convergent, the sum is 1 . It is com the sum is " e. None of these 7. Find the common ratio of the geometric series a 1e b.e ce d.e e. None of these Determine whether the series Σ a. It is convergent 9. Determine whether the series is convergent or divergent. b. It is divergent n +3 is convergent or divergent. 7n a .It is convergent b.It is divergent 10. Find a formula for the general term a, of the sequence, assuming that the pattern of the terms continues. 1 4 2 16 25 2'3'4' 5 6 a. b. 25 n d. -.- e. None of these

taotion First think geometric series 4. Use substitution in a known series to find the Maclaurin's Series for the following functions. Be sure to include the interval of convergence in your answer. (a) f(z) =- (b) f(z) = ln(1-H) (c) f(z) = sin(2x) (d) f(z)=ea-2 5. Use term-by-term differentiation or integration of a known series to find a power series rep- resentation for each of the following functions. Be sure to include the interval of convergence in your answer. (a) f(z)=2cos(2x) (b) f(z) (c) f(z)=arctan(3x) (d) f(x)-In(1 +2x) 6x5 (e) f(z) = (14 6. Express the given parameterizations in the form y- f(x) by eliminating the parameter. (c) r(t) = ln(t), y(t)-3t + 2 (a) z(t) = 3e, y(t) = 7c" + 1 (b) () 6+t1, (t) 32 7. Find parametric equations for the segment joining the points P(L, 2) and Q(8,-4). 8. Use the formula for the slope of the tangent line to find for the curves: i. c(t) = (5e,22-7) at t-1. ti. c(t) = (3e, 2t + 2e-t) at t = 0.