Chapter 1 Flashcards
(x→0)lim(sinx/x)
1
(x→0)lim((1-cosx)/x))
0
T or F: ∞/∞ is a valid answer
F; that’s called indeterminate form
terms for “No Answer” for functions & limits
Undefined for functions; DNE for limits
how to find VA
literally just set x equal to whatever number will render the function undefined (which is usually 0, for stuff like ln, and when it’s an expression in the denominator)
3 steps to guaranteeing c can be found w/ Intermediate Value Theorem
1) Sketch the graph within the interval
2) see if f(intvl val to left) and f(intvl val to right) exist and surround the value of c (which is prob 0)
3) Declare whether IVT can be used, and use trace to find c on that interval
what equation do you need to remember when finding the derivative
f¹(x) = (h→0)lim(f(x+h) - f(x))/h
2 steps to find tangent line if you have derivative & point of tangency
1) plug the x-val of the given point into derivative to get the slope of the tangent line
2) use y - y1 = m(x - x1) to get the tan line equation by plugging in the newly found slope & coordinates of the given point; simplify to get the line
how to get 1 from sin & cos
1 = sin² + cos²
how to get tan from sin & cos
tan = sin/cos
sin2θ
2sincos
cos2θ with sin & cos
cos² - sin²
cos2θ with sin
1 - 2sin²
cos2θ with cos
2cos² - 1
what are you looking for when solving limits where x reaches positive/negative infinity
horizontal asymptotes
HA when n deg > d deg
NA
HA when n deg < d deg
0
when x reaches negative infinity for a limit, what is the sqrt(9x²)?
-3x
3 steps to find continuity
1) Is f(x) defined?
2) Is the limit that f(x) when n -> x defined?
3) Do those values equal each other?
If all 3 are right then it’s continuous
How to find if a discontinuity is removable or not thru a limit
Take the limit at x reaching the discontinuity:
- Get # back: removable
- Get DNE back: non-removable
How to find if a discontinuity is removable or not thru a graph
- Holes are removable
- Breaks/VA’s are not
How to make function continuous (really just 1 step)
1) Make a limit a little less than the number that breaks the piecewise function, and a limit a little more
2) Set them equal to each other and solve
What is the derivative for purposes to answer that dang question 15
if x in f’(x) is the x-val of the point of tangency, then plugging it in to the derivative equation gives the slope needed to touch this point in the line of tangency.
If you need the y-coord for the point of tangency, take the x-tan-coord & plug into original f(x) that the derivative came from
