Derivatives
-sin(x)
What is the derivative of cos(x)?
draw/write down what you know and what you need.
What is the first step to solve an optiminzation problem?
how the outputs of a function behave as inputs approach some x value, c.
What is a limit?
local max, local min, absolute max, absolute min
What are the types of extrema?
a circle often used in trigonometry which displays values such as
1/2 and sqrt(2)/2 and sqrt(3)/2
What is the Unit Circle?
lim_(h->0)(f(x+h)-f(x))/h
lim_(h->0)(f(a+h)-f(a))/h
lim_(x->c)(f(x)-f(c))/(x-c)
What is the limit definition of the derivative?
the objective function and constraint
What two elements do we use to solve an optimization problem?
lim_(x->c^-)f(x)=lim_(x->c^+)f(x)=f(c)
What is the limit definition of continuity?
a point in the domain of a function when f'(c) equals 0 or DNE.
What is a critical point?
(ADJACENT)/(HYPOTENUSE)
What is cotangent?
derivative of the first times the second, times derivative of the second.
dy/dx = dy/(du) * (du)/dx
What is the Chain Rule?
What does the objective function tell us?
if f(x) is a continous function on [a,b], then for every d between f(a) and f(b), there exists a c between a and b so that f(c)=d
What is the IVT?
find the critical points and place them with the values of the closed interval and plug them into the original function.
How to make a candidate table.
Sin(θ) = Opposite / Hypotenuse. Cos(θ) = Adjacent / Hypotenuse. Tan(θ) = Opposite / Adjacent
What is SOHCAHTOA?
1/(xlna)
What is the derivative of logax?
Answer the question. Include units if possible.
What is the last step in optimization?
What is the first step in finding a limit?
if f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then there is at least one point c in (a,b) at which
f'(c)=(f(b)-f(a))/(b-a)
What is the Mean Value Theorem?
sin2x+cos2x=1
1+tan2x=sec2x
1+cot2x=csc2x
What are the Pythagorean Identities?
-(1)/(|x| sqrt(1-x^2)
What is the derivative of csc-1(x)?
Looks for the smallest or largest value.
What is Optimization?
concerns the limit of a function that is squeezed between two other functions each of which has the same limit as a given x value.
What is the Squeeze Theorem?
a function that is always increasing or decreasing on an interval.
What is a monotonic function?
sin2x=2sinxcosx
cos2x=cos2x-sin2x
What are Double Angle Identites?