Limits
Describe the end behavior of f(x) = ex from left to right.
limx→∞ f(x) = 0
and
limx→2 f(x) = ∞
What is velocity?
Speed with direction.
What is ∫sinxdx equal to?
∫sinxdx = -cosx + c
What is the formula for the chain rule?
f(x) = g(h(x))
f(x) = g'(h(x))*h'(x)
What is ∫f(t)dt on the closed interval of [a,a]?
= 0
What is the notation of the limit from the right and left?
limx→a+ f(x) and limx→a- f(x)
What is the Intermediate Value Theorem ( IVT )?
If a function is continuous on the closed interval of [a,b], the f(x) contains every y-value between f(a) and f(b).
What is sin(45°)?
= sqrt2 / 2
What is the formula to find volume using the Washer Method?
V = π∫[R(x)2 - (r(x))2]dx
What method would you use to solve this equation?
U-Substitution!
What must be true for the limit to exist?
The limit from the left and right must equal each other.
What is a derivative?
A derivative is a function, f'(x), that gives the slope of the curve at any x-value on f(x).
What is something in common between the derivatives of cos, cot, and csc of θ ?
They are all negative.
What is the formula for the average value of a function?
c* = (1/(b-a))∫f(x)dx on the closed interval of [a,b].
What is ∫(ax)dx equal to?
= (1/lna)*ax + c
What is the indeterminant form?
When a limit is evaluated and the answer comes out to be 0/0 or ∞/∞, meaning there is a way to alter said function in order to properly evaluate the limit.
What are tangent lines and what are they used for?
Tangent lines are lines that touch a curve at a point, matching the curve's slope there. They can be used to approximate function values near the point of tangency.
What is sec(90°)?
Not defined
What is the formula for the volume using the disk method?
V = ∫πr2dx
OR
V = π∫r2dx
How would you rewrite ∫f(t)dt on the interval of [b,a] if you wanted it on the interval of [a,b] ?
- ∫f(t)dt on the interval of [a,b]
What is the only discontinuity that a limit exists as a definite number?
A hole or removable discontinuity.
What is implicit differentiation / when is it necessary?
Implicit differentiation is necessary when an implicit function is present, which where the dependent variable (y) is not isolated on the side of the equation.
example: x2+xy-y2 = 1
What is d/dx (arcsec(x)) equal to?
= (1) / (|x|*sqrt(x2 - 1))
What is the formula for the quotient rule ( f(x)/g(x) )?
= (g(x)(f'(x))-f(x)(g'(x))) / g2(x)
How do you find the final position, s(b), using the initial position, s(a), and displacement?
s(b) = s(a) + ∫v(t)dt on the interval of [a,b]