Trig
The range of cos(x)
What is is [-1,1]?
The derivative of position is
Velocity
The limit as x approaches 0 from the right of ln(x).
What is negative infinity.
You should use what to take the derivative of x^2 sin(y) + y sin(x) = 1
Implicit differentiation
Given a function f(x), 2f(x-3) does this to the graph of f(x)
What is a horizontal shift by 3 to the right and a vertical stretch by a factor of 2?
If tan(x) = 3 where x is in [0,pi/2], sin(x) is.
What is 3/sqrt(10)?
The slope of the tangent line to y = 2e^x + x at the point (0,2) is
What should you do to find the limit as x approaches 0 of (8^x - 5^x)/x?
This is an indeterminate form of type 0/0 so you should use L'Hopital's rule.
If you want to find all the local mins and maxes of a function f(x), what should you do?
Take the derivative, find all the critical points. Then build your number line to check where f is increasing/decreasing around the critical points.
What is the derivative of b^x? Where b is a constant.
ln(b) b^x
What is the most general antiderivative of e^3 + 2sin(x)?
xe^3 - 2 cos(x) + C
What is the limit definition of the derivative?
f'(x) = limit as h approaches 0 of
(f(x+h) - f(x))/h
What type of indeterminate is limit as x approaches infinity of x sin(pi/x)?
Indeterminate product
Name the key steps you should do when approaching a related rates problem.
What do you know? What do you want to know? Draw a picture. Write down an equation relating your unknowns. Implicitly differentiate.
There is a horiztonal asymptote y = 2/3.
The derivative of tan(x) is
sec(x)^2
The quotient rule says that (f/g)' is
(f'g - g'f)/g^2
The limit as x approach a of f(x) equals f(a).
What type of differentiation do you need to find the derivative of x^(tan(x))?
Logarithmic differentiation
Find the linearization of f(x) = sin(x) centered at a = 0.
L(x) = f(a) + f'(a) (x-a) = 0 + x = x
What type of symmetry does sin(x) have?
It's periodic and odd.
The chain rule says (f(g(x))' =
f'(g(x)) g'(x)
Show that the function
f(x) =
x^2 + 1 when x< 0
e^x when x >= 0
is continuous.
Check that the limit as x-> 0 from the left and the right is 1. This equals the functions value at 0, which is e^0 = 1.
What should you use to show that if f'(x) = g'(x) on an interval then f(x) = g(x) + d for some constant d.
Mean value theorem.
Name the key steps you should take when doing an optimization problem?
Draw a picture. Write down an equation for what you want to optimize in terms of your unknowns. If there are multiple unknowns, relate them using a different equation. Substitute this in and take the derivative of what you want to optimize. Check the critical points (and possibly endpoints) to find which one is the max/min you seek.