Find the remainder in the Taylor series centered at the point a for the following function. Then show that lim |Rn(x)=0 f(x)=ex f(x) = e a=0 n-∞ First find a formula for f (n) (x). f(n) (x) = (Type an exact answer.) Next, write the formula for the remainder. n+1 Rn(x) = (n+1)! for some value c between x and 0 = 0 for all x in the interval of convergence. (Type exact answers.) Find a bound for Rn(x) that does not depend on c, and thus holds for all n. Choose the correct answer below. ex elx OC. R(x)(n+1 OE. Rn(x)(n+1) | Rn (x)| = (n+1)* = 0 for all x in the interval of convergence by taking the limit of the bound from above and using limit rules. Choose the correct reasoning below. Show that lim R,(x)=0 OA. Use the fact that lim U = 0 for all x to obtain lim |R,(x)| = el*1.0=0. OB. Use the fact that lim = 0 for all x to obtain lim |R,(x)=1+0=0. OC. Use the fact that lim A(+1) (n+1)! = 0 for all x to obtain lim R₁(x) =+0=0. e OD. Use the fact that lim = 0 for all x to obtain fim R₁(x)| =…

Consider the following parametric equations, x=-4t, y=-7t+ 13; -10 sts 10. Complete parts (a) through (d) below. a. Make a brief table of values of t, x, and y t x(t) y(t) 10 -6 0 6 10 (Type integers or decimals.) ○ A. b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t). 130 G c. Eliminate the parameter to obtain an equation in x and y. d. Describe the curve. OA. A line segment falls from left to right as t increases OB. A line segment falls from right to left as t increases OC. A line segment rises from right to left as t increases OD. A line segment rises from left to right as t increases

Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y-axis. -1 y=10 (1+10x) 1 y= 0, x = 0, and x=2 Set up the integral that gives the volume of the solid using the shell method. Use increasing limits of integration. Select the correct choice and fill in the answer boxes to complete your choice. (Type exact answers.) OA. S dx O B. dy The volume is (Type an exact answer.)

Find the slope of the line tangent to the following polar curve at the given point. r = 1 - sin 0; Find the slope of the line tangent to the polar curve at the given point. Select the correct choice below and, if necessary, fill in the answer box within your choice. OA. The slope of the line tangent to the polar curve at the point OB. The slope of the line tangent to the polar curve at the point (2) 1 元 (1) 6 is (Type an exact answer.) is undefined.

Determine whether the following series converges. 4(-1)k Σ k=0 3k+6 Let a > 0 represent the magnitude of the terms of the given series. Select the correct choice below and fill in the answer box(es) to complete your choice. OA. The series diverges because ak is nonincreasing in magnitude for k greater than some index N and lim ak koo B. The series converges because ak is nondecreasing in magnitude for k greater than some index N. OC. The series converges because ak OD. The series diverges because a₁ = OE. The series converges because ak ak and for any index N. there are some values of k > N for which ak+1 ≥ak and some values of k > N for which ak+1 ≤ak- is nondecreasing in magnitude for k greater than some index N is nonincreasing in magnitude for k greater than some index N and lim ak K-00 OF. The series diverges because a₁ = and for any index N, there are some values of k > N for which ak+12 ak and some values of k > N for which ak+1 sak-

1 × 1016 1 × 1041 = 1 × 10? = ? = product's exponent Part 2 (0.5 point) 1 × 1023 × 1 × 1021 × 1 × 1020 1 × 1015 × 1 × 1021 = 1 × 10? = ? = = product's exponent

K A differential equation and its direction field are given. Sketch a graph of the solution that results with each initial condition. 2 y'(t) = 2 y(-1)=-2 and y(-2) = -1 y +1 Which of the following shows the solution that results with the initial condition y(-1)=-2? O A. J +21 Which of the following shows the solution that results with the initial condition y(-2)=-1? ○ A. +2

4t Does the function y(t) = 6e satisfy the initial value problem y(t) - 4y(t) = 0, y(0)=5? Choose the correct answer. A. Yes, it satisfies the initial value problem. This is because it satisfies the differential equation OB. No, it does not satisfy the initial value problem. This is because it satisfies the differential equation but does not also satisfy the initial condition. OC. Yes, it satisfies the initial value problem. This is because it satisfies the initial condition. OD. No, it does not satisfy the initial value problem. This is because it does not satisfy the differential equation. OE. Yes, it satisfies the initial value problem. This is because it satisfies the differential equation and also satisfies the initial condition.

K Determine whether the following series converges. Justify your answer. 5 10k + k Σ 5 k=1 5k -2 5k-2 Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) OA. The series is a p-series with p= so the series diverges by the properties of a p-series. so the series converges by the Ratio Test. OB. The Ratio Test yields r = O C. The limit of the terms of the series is OD. The series is a p-series with p= so the series diverges by the Divergence Test. so the series converges by the properties of a p-series. OE. The series is a geometric series with common ratio so the series diverges by the properties of a geometric series. OF. The Root Test yields p = . so the series converges by the Root Test.