Limits
lim as x approaches 0 of sin(x)/x
1
What is the formal limit definition of a derivative?
limit as h approaches 0= (f(x+h)-f(x))/h
Anti derivative of csc2(x)
Mean Value Theorem
If a function f is continuous on [a,b] and differentiable on (a,b), then Mean Value Theorem says there exists a value c on the interval (a,b) such that f'(c)= Average Rate of Change over (a,b).
f(x)=4x3-3x2. At which x value(s), if any, does f have a point of inflection?
f has a point of inflection at x=1/4
lim as x approaches zero from the right = 4
lim as x approaches zero from the left = 4
f(0)= 12
What type of discontinuity does f have at x=0?
Removable discontinuity
Find the derivative: x12-10x4+e1,256291
12x11-40x3
Anti derivative of 1/x1/2
2x1/2
For a twice differentiable function f, f(1)=5 and f(10)=27. Must there be a value c such that f(c)= 12? Explain using the correct theorem.
Since f is twice differentiable, it is therefore continuous on [1,10] and differentiable on (1,10). Since f(1) does NOT equal f(10), Intermediate value theorem says there exists a value c such that f(c)= 12.
The side lengths of a cube are increasing at a rate of 6 cm/sec when the side length= 2 cm. At what rate is the area changing when s=2?
144 cm2/sec
lim as x approaches 2 of (x2-4)/(x-2)
4
What is the derivative of sec2(x)?
2sec2(x)tan(x)
Given f(x)= (4(x2+1))/(x2), what is the average value from [1,3]?
x= 31/2
lim as x approaches 1 of (ex-1-1)/lnx
Since lim as x approaches 1 of ex-1-1=0 and lim as x approaches 1 of lnx=0, by L'Hopital's, lim as x approaches 1 of (ex-1-1)/lnx= lim as x approaches 1 of ex-1/(1/x). Therefore lim as x approaches 1 of (ex-1-1)/lnx= 1.
A particle's position is given by s(t)= cos(t+2)/(t+2) for t is greater than or equal to 0.
At t=5 seconds is the particle speeding up or slowing down?
Speeding Up
v= -0.1092 meters/sec
a= -0.03
lim as x approaches 2 FROM THE RIGHT of 1/(x-2)2
infinity
Find the derivative of f(x)= (ln(x2+4)1/2)/x
x/(x2+4)-(1/x)
Find the anti derivative of 3x2-4x+5
x3-2x2+5x+c
What is the mean value theorem for integrals.
If f is continuous on [a,b] then there exists a value c in the interval [a,b] such that f'(c)= average value over [a,b].
Water drains from a cylindrical tank with a radius of 4 cm. The height is decreasing at a rate of 0.5 cm/sec. How is the rate of water in the tank changing when h=6?
-8 pi
lim as x approaches -5 of (sin(x+5))/(x2+7x+10)
-1/3
Second derivative of cot(4x)
32csc2(4x)cot(4x)
f''(x)=6, f(1)=12, and f(-2)= 18. Find an explicit equation for f(x).
f(x)= 3x2+x+8
Use squeeze theorem to find the limit as x approaches zero sin(x)/x.
limit as x approaches zero sin(x)/x = 1
f(x)= x3-(3/2)x2
Find all absolute maxes on [-1,2].
f reaches an absolute max of 2 where x=2 and an absolute max of x= -5/2 where x=-1.