Definitions/Theorems
What is a function?
A rule that assigns a unique output value for each input value.
Evaluate:
limx→1 x2 +1
2
Find the derivative of the following function:
y = ex - (1/x)
y' = ex + (1/x2)
∫0π cos x dx
sin π - sin 0 = 0
Find the formula for the nth term of the sequence:
2, 5, 10, 17, 26, 37,...
an = n2 + 1
What three conditions must be met for a function f(x) to be continuous at a point x=c?
1) f(c) exists
2) limx→c f(x) exists
3) limx→c f(x) = f(c)
Let f(x) = 3x2+1 and g(x) = 5x2-4. Evaluate limx→∞ f(x)/g(x) .
3/5
Find the derivative of the following function:
y = x2 cos(x)
y' = 2x cos(x) - x2 sin(x)
Approximate ∫02 5x4 dx using Simpson's Rule with n=4.
32 1/12
Determine if the series converges or diverges and state test used:
∑n=0∞ ln (1/n)
Diverges by nth term test.
State the quotient rule for derivatives for functions f(x)/g(x).
(gf'-fg')/g2
Evaluate:
limx→0+ ln(x+1) / x
1
Find the second derivative of the following function:
y = cos x - 1/x
y'' = -cos x - 2/x3
Find the area between y=2-x2 and y=-x.
With bounds -1 to 2, the area is 9/2.
Determine if the series converges or diverges and state test used:
∑n=0∞ -1/n2
The series converges by p-series and ACT.
Daily Double!
State the Intermediate Value Property/Theorem.
If f(x) is a continuous function, then whenever f(x) takes on two values, it must also take on all of the values in between.
Evaluate:
limx→0 (sin x2 ) / x
0
Find the derivative of the following function:
y = ln(sinh x)
y' = coth x
Find the volume of the solid generated by revolving the region bounded by y=2x-1, y=√x, and x=0 about the y-axis.
7π/15
Determine if the series converges or diverges and state test used:
∑n=0∞ (2n)!/n!n!
The series diverges by the ratio test
State the Alternating Series Test.
An alternating series converges if
1) un are all positive
2) un are eventually non-increasing
3) un → 0
Daily Double!
Evaluate:
limx→∞ (ln 2x - ln(x+1))
ln 2
Use implicit differentiation to find the derivative of the following function:
2xy + y2 = x + y
y' = (1-2y)/(2x+2y-1)
∫ sech2(x - 1/2) dx
tanh(x-1/2)+c
For what values of x does the series converge:
∑n=0∞ (x-2)n / 10n
x between -8 and 12