Definitions
Limits
Continuity
Differentiability
100
What is the definition of a limit?
A limit is a value that a function approaches but does not pass
100
What are the three ways of finding limits?
Numerically, graphically, and analytically.
100
When is a function continuous at a point?
When when f(x) as x approaches c is the same as f(c).
100
Does a function have to be continuous to be differentiable?
Yes
200
What is continuity?
When a function is one line or curve on its graph
200
Find the limit of f(x) = tan(x) / tan(2x) numerically.
x: -.1, -.01, -.001, .001, .01, .1 f(x): .494, .49995, .49999, .49999, .49995, 4.94
200
Can polynomials ever not be continuous? If so, give an example.
Polynomials are always continuous.
200
What is the three step definition of continuity?
1. The function is defined at a point. 2. The function has a limit as it approaches the point. 3. The limit and the function at the point are the same.
300
What is the definition of a derivative?
A rate of change
300
If the limit approaching b of f(x) is 7 and the limit approaching b of g(x) is -3, what is the limit of (f(x) * g(x)) as x approaches b?
lim f(x) * g(x) = lim f(x) * lim g(x) 7 * -3 = -21
300
Is the piecewise function f(x) = { 2x^2 - 2 for x<=2 } { 5x - 4 for x> 2 } continuous?
Yes, the limit of both functions as x approaches 2 is 6, so it is continuous at that point, and as both functions re polynomials there are no other points that could be discontinuous.
300
What is the four step definition of differentiability?
1. Find what the function equals at a point 2. Find the limit of the function 3. Prove that the limit and function is the same at the point 4. Prove the slope is the same on both sides.
400
What is the formula for derivative at a point?
[f(x) - f(c)] / (x- c)
400
Can a function have a limit at a point where the function has no value?
Yes, if both sides leading up to that point have a limit at the same value.
400
What is the limit of (3 - x) / (x^2 - 9)?
divide out (x-3) and you are left with -1 / (x+ 3). From there, plug in 3 for x and get 1/6.
400
On this graph, give a point where the function isn't differentiable, and a reason why.
-1, the slopes are different for each side.
500
What is local linearity?
As you zoom in on a point, the function starts to look like a linear function whose slope is the same as the derivative at the point.
500
What is the limit from the right and the limit from the left of f(x) = sec(πx / 4) as x approaches -2? You can use a graphing utility.
-infinity from the left, infinity from the right.
500
What is the limit of this function as x approaches 3?
Rationalize the numerator, divide out (x - 3), plug in for x equaling (1/4).
500
Use the 4 step definition of differentiability to find whether f(x) = { x^2 + 1 x=<2 } { 4x - 3 x >2 } is differentiable or not at x = 2
1. f(2) = 5 2. lim x -> 2 f(x) = 5 3. f(x) and the limit of f(x) are the same 4. The slopes are the same at 4
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