### Finding antiderivatives

Antiderivative of: (t5 + 6t3) dt

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Antiderivative of: (t5 + 6t3) dt

See attachment

I attached a word document. Be sure to show me all of your work so that I can fully understand how to do the problems correctly. Thank you very much for your help.

Show that the sum from 0 to infinity of (1-x)x^n does not converge uniformly on [0,1]. What subintervals of [0,1] does it converge uniforlmly on?

What is the fifth term of the series an = n + an-1' if a0 = -3.

Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae. Describe the importance of each example and why a linear model is appropriate for the data. Note that we are referring to a linear model not simply a time chart where dots

Write out the Riemann Sum for R(f,P, 0, 2) for arbitrary n, f(x) = x2−3x+2, where each ∆xk = 2/n and ck = xk, simplify and use the formulas ∑n,k=1 k=(n(n+1))/2 and ∑n,k=1 k2=n((n + 1)(2n + 1))/6 to find the limit as n --> 1.

See attached lim (sin^2 3t)/2t t--->0

Please see the attached file for the fully formatted problem. Lim (1 - cos t)/2t t--> 0

See attached

Evaluate Limit[tan (a*theta)/sin(b*theta)] as theta approaches zero.

Calculate limit (1/x^2) as x approaches infinity

Prove that Lim (3x)=6 as x approaches 2 using Epsilon-Delta definition.

Define a limit graphically

Determine whether the series Sum(n!/n^n) n=1..infinity is absolutely convergent, conditionally convergent or divergent.

Find lim (ln x)^3/(x) as x--> infinity

Test the series (in the attached file) for convergence or divergence by using the Comparison Test or the Limit Comparison Test.

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

The question is in attached file. Suppose a sequence of continuous functions, { ?n }, has the property that ?n  ? and  > 0,   > 0 such that if | x - y | <  then n, | ?n (x) - ?n (y)| <  Prove that ? is continuous.

Questions are in attached files. Thank you.

Summation Series. See attached file for full problem description.

I have managed to evaluate the double integral using the horizontal simple method, and answer 63. But when I reverse the order (vertical simple method) I cannot reach the same answer of 63, I get 65. 1) Evaluate the double integral of f(x,y)=x+4y^2 over the triangular region with vertices (-2,2) (4,2) & (1,-1) Check that r

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

I need a proof for "If f on [a,b] is continuous & 0 is not a member f([a,b]) then f is bounded away from 0."

For numbers a1,....,an, define p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that: (a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0 Prove that there is some point x in the interval (0,1) such that p(x) = 0

Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b].

Prove that if f : [a,b] ----> R is a bounded function that is continuous at all but finitely many points, then f is integrable over [a,b].

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is.

Suppose that the function F:R->R has derivatives of all orders and that: F"(x) - F'(x) - F(x) = 0 for all x F(0)=1 and F'(0)=1 Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.

Please see the attached file for the fully formatted problems. Suppose ak 0 and a1/k k ! a as k ! 1. Prove that P 1k =1 akxk converges absolutely for all |x| < 1/a if a 6= 0 and for all x 2 R if a = 0. Since this problem is an analysis problem, please be sure to be rigorous.