True/False
If the terms of an infinite series converge to 0,
Then the sum converges to some number L.
False, 1/n is a counter example.
What is the limn->23x2+3.
3(2)2+3=12+3=15
If there is a series with terms of the form 1/np for n=1,2,3,..., when does it converge?
When p>1
f(x)=x*cos(x)+3x-1, find f'(x).
f'(x)=-x*sin(x)+cos(x)-3x-2
Let f(x)=3x2+cos(x)+x-1, find F(x).
F(x)=x3+sin(x)+ln(x)+C
If the absolute value of an+1 divided by an equals 1,
Then the series diverges.
The ratio test is inconclusive when equal to 1.
What is the limx->oo(6x3+1)/(2x3+x+3)
6/2=3
What is the sum of the series with terms equal to 3/2n for n=2,3,4,...?
This is geometric with r=1/2 and the first term being equal to 3/4. Using the geometric sum formula we get (3/4)/(1-(1/2))=3/2
Give an example of 2 functions, from different families of functions, that are continuous at x=0 but not differentiable at x=0.
f(x)=(x2)1/2 and g(x)=x2/3
What does the integral from a to b of a function f(x) represent?
The net area between f(x) and the x-axis. Not just the area.
If the sequence of partial sums of a series has an upper and lower bound,
Then the series converges.
False, (-1)^n has a partial sum in between -2 and 2 but does not converge.
What is the limx->0sin(x)/tan(x)
sin(x)/tan(x)=sin(x)/sin(x)/cos(x)=cos(x), cos(0)=1
What are the conditions for convergence using the integral test?
Let an=f(n). For all x greater than some k we have:
1. f(x)>0.
2. f(x) is continuous.
3. f(x) is decreasing.
Let f(x) be a piecewise function defined by f(x)=x2sin(1/x) for x not equal to 0 and f(x)=0 for x=0. Is f(x) differentiable at x=0? Justify.
f'(0)=limh->0(f(h)-f(0))/h=(h2sin(1/h))/h=hsin(1/h),
sin(1/h) is always less than 1 so we can say -h<=hsin(1/h)<=h, and h=0 so hsin(1/h)=0 and f'(0)=0.
Let F(x) be equal to the integral from 2 to x of f(t)dt. If f(5)=0 and f(t) changes from positive to negative at t=5, what can you say about F(x) at x=5?
F(x) has a local maximum.
If the sum of ans converges for n=1,2,3,...,
Then the sum of an2s converges for n=1,2,3,....
False, if the series of ans is conditionally convergent, the sum of an2s may not converge. The alternating sum of square roots is a counter example.
What is the limn->oox100000/ex
Exponentials always dominate polynomials so the limit equals 0. We can use L'Hopital's Rule over and over again to get a constant over ex.
What is the sum of all (n*ln(n))-1 for n=2,3,4,...?
Use the integral test to see that it diverges.
Let f be the function such that f(2)=5 and f'(2)=-3. Use a linear approximation to estimate f(2.1).
f(x+h)=f(x)+hf'(x) so
f(2.1)=5+.1*(-3)=4.7.
Let F(x) be equal to the integral from 0 to x of (t2-4t)dt. Find all values of x for which F(x) is increasing and concave down at the same time.
F'(x)=x2-4x=x(x-4), so F(x) is increasing when x>4 or x<0. F''(x)=2x-4=2(x-2) so F(x) is concave down for x<2. So F(x) is both concave down and increasing when x<0.
If the integral from 0 to infinity of a function f(x) converges,
Then the sum of f(x) for x=1,2,3,... also converges.
If f(x) is not decreasing then the integral test may not be used.
What is the limx->0(ex-1-x)/x2
Use the Taylor polynomial for ex to get:
(1+x+(x2/2)+(x3/6)+...-1-x)/x2=1/2+x/6+...
Then evaluate limit to get 1/2.
Determine if the sum of all (-1)n(n*ln(n))-1 for n=2,3,4,... is absolutely convergent, conditionally convergent, or divergent.
Conditionally convergent.
Let f(x) be twice differentiable and let:
f(0)=0, f'(0)=0 and f''(0)>0 for all x. Justify why f(x)>0 for all x not equal to 0.
Since f''(x)>0 we know f'(x) is always increasing. Therefore when x<0 f'(x)<0 and when x>0 f'(x)>0. When x>0 f(x) is increasing thus f(x)>0, when x<0 the function is decreasing towards 0 thus f(x)>0 for all x not equal to 0.
Let F(x) be the integral from 1 to x of (t3-6t2+9t)dt. Find the absolute maximum on the interval [0,4].
F'(x)=x3-6x2+9x=x(x-3)2, so F'(x)=0 if x=3,0.
x(x-3)2 is always nonnegative if x>0 so we just check the end points 0 and 4. F(0)=0, F(4)=21/4.