Limits & Derivatives Flashcards
Why are limits the foundation of calculus?
Algebra fails when calculating slope of a line in a specific instance. Need calc it as it approaches (limit) of that numbers
How do you solve tangent and velocity problems?
Find limit of the slope as x gets closer and closer to zero. That gets the slope of the tangent line, which is the velocity / instantaneous rate of change
What are the limit laws for + - x /
Because limit(f) is just some number. So it’ll add, divide, etc
Limit as x goes to a of x =
a
Intuitive definition of the limit, and more precise definitions
intuitive: what happens to a function when you get close to a point
Precise:
left hand limit and right hand limit exist and are the same, then the limit exists
What is the squeeze theorem?
What 3 things does a Continuous function have?
(Must remember this) Esp #3
What are types of non-continuous functions?
Removable (pick up pencil)
Jump ([[x]]
Is it true that all polynomials and rational functions are continuous?
yes
What does the screenshotted theorum mean?
If the outside function is continuous, then you can bring the limit inside…
if outside function is continuous, you can bring the limit inside
The classic, know this one
What is intermediate value theorum?
Must be continuous and closed interval
Is there a number exactly one more than its cube?
Due to intermediate value theorem, know that all polynomials cross x axis at least once, so there is a solution to that equation equals zero
Do Limit Laws apply to infinite limits?
No, because infinity is not a number
Whats infinity minus infinity?
DNE because infinity is not a number
notice the method of multiplying by (1/x3) / (1/x3)
Ratio of leading coefficients.
HARD problem
multiply by conjugates
Try to simplify getting numerator to 1
Can distribute multiplication into the square root.
Tricky one.
USe squeeze theorem - think that when you see cos
The equation of the derivative of a function f at a number a is….
limit as h goes to 0
Write equation describing that the tangent line to y = f(x) at (a, f(x))
Y - f(a) = f’(a) (x - a)
Y changes as the rate of derivative times change in x
write limit every time
Write the initial formula every time
Whats the difference between average velocity and instantaneous velocity?
Average is given over two points
Instantaneous is a limit as h –> 0
Are these all the same thing?
slope of tangent line
instantaneous rate of change
instantaneous velocity
marginal rate of change
derivative
yup
T/F: If a function is differentiable, it is it continuous.
True. Differentiable means it has a derivative. And thus it is continuous.
The opposite is false though, it can be continuous but not differentiable
Name some situations where a function is not differentiable
Discontinuous or has a corner
|x|
Has vertical asymptote
[[x]]
How do you get the second derivative? And what does it mean?
Derivative of the first derivative.
It’s the rate of change of the rate of change. e.g. acceleration.
What is third derivative called?
The jerk.
a large jerk means a sudden change in acceleration.
If you see that subtraction at bottom, think of this formula - f(x) - f(a) / x-a
Split into limit from left and right, piecewise
What is difference of cubes simplification?
b3 - a3 = ?
(b-a)(b2 + ab + b2)
For a function, how do you go about finding the horizontal and vertical asymptotes?
Horizonal: look at x –> infinity and negative infinity
Vertical: look at zeros of denominator. Then if you cannot factor that out, its a VA
When have continuous function, the limit can pass through…
What are various methods of calculating limits
Limit Laws
Direct substitution
Squeeze theorem
Conjugate method
Factoring
Limit translates inside lim f(g(x)) = f(lim g(x)) …. if f is continuous
