Algebra/Trig
Factor 4x2 -4x - 3
(2x - 3)(2x + 1)
What is chosen first, the epsilon or the delta?
The epsilon.
When are you allowed to use L'Hopital's rule?
When the limit is indeterminant. 0/0 or infinity/infinity
Find the second derivative of the following
y = 9 tan(x/3)
y'' = 2sec(x/3) sec(x/3) tan(x/3)
Using epsilons and deltas, describe when a number L is not the limit of f(x).
When we choose an epsilon so that there does not exist a corresponding delta so that
|x - x0| < delta --> |f(x) - L | < epsilon
What is the tangent at the angle 765o ?
1
Given f(x) = x, and epsilon = 1, find a delta to satisfy the epsilon/delta definition for the limit as x approaches 2.
delta = 1
Find the limit as x -> infinity of the following
f(x) = (3x-2) / (1 - 7x)
-3/7
Find the 65th derivative of y = e-2x
-265 e-2x
Why does sin(1/x) have no limit as x approaches 0?
Oscillating
Find the horizontal and vertical asymptotes of:
( x - 7 ) / (x2 - 3x + 2)
HA: y = 0
VA: x = 2, x = 1
delta = epsilon/5
What are the 3 requirements for the function to be continuous at a point x0?
The function must be defined, the limit must exist, and the limit must equal f(x0).
What is the derivative (with respect to x) of au, where u is a function of x?
(d/dx) au = au ln(a) (du/dx)
What is the intermediate value theorem and how is it used?
We must have a continuous function on an interval [a, b] in the x direction that maps to [f(a), f(b)]. If y is in the interval [f(a), f(b)], then there must exist an x so that f(x) = y.
We can use it to show when functions do and don't cross the x-axis (aka have roots).
State the 3 Pythagorean identities.
sin2x + cos2x = 1
1 + cot2x = csc2x
tan2x + 1 = sec2x
Given epsilon > 0, and f(x) = k, what do we choose delta to be as x approaches any x0?
delta can be anything because the constant function f(x) = k gives us a true statement. k - k = 0 < epsilon, is always true.
Find the limit as x -> 0 of
(x - sin x) / x^3
1/6
Find the derivative
y = exp(sec2(tan(2x)))
exp(sec2(tan(2x))) (2sec2x)(tan(x)tan(2x) + sec2(2x))
Find the limit as x -> 0 of
(x + x cos(x)) / (sin (x) cos(x))
1
Which trigonometric functions are even, and which are odd?
Cosine and Secant are even, the rest are odd.
What is the relationship between limits, continuity, and derivatives?
all differentiable functions are continuous, all continuous functions have a limit.
Find the limit as x -> 0 of
(ex - 1) / (cos(x) - 1)
DNE
Find the derivative
y = xsin(2x)
xsin(2x) (2cos(2x) ln(x) + sin(2x)/x)
Find the limit as x -> infinity of
(1 - 1/x)^x
1/e