Limits & Derivatives

Question 1

Why are limits the foundation of calculus?

Answer 1

Algebra fails when calculating slope of a line in a specific instance. Need calc it as it approaches (limit) of that numbers

Question 2

How do you solve tangent and velocity problems?

Answer 2

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

Question 3

What are the limit laws for + - x /

Answer 3

Because limit(f) is just some number. So it’ll add, divide, etc

Question 4

Limit as x goes to a of x =

Answer 4

a

Question 5

Intuitive definition of the limit, and more precise definitions

Answer 5

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

Question 6

What is the squeeze theorem?

Answer 6

Question 7

What 3 things does a Continuous function have?

Answer 7

(Must remember this) Esp #3

Question 8

What are types of non-continuous functions?

Answer 8

Removable (pick up pencil)
Jump ([[x]]

Question 9

Is it true that all polynomials and rational functions are continuous?

Answer 9

yes

Question 10

What does the screenshotted theorum mean?

Answer 10

If the outside function is continuous, then you can bring the limit inside…

Question 11

Answer 11

if outside function is continuous, you can bring the limit inside

Question 12

The classic, know this one

Answer 12

Question 13

What is intermediate value theorum?

Answer 13

Must be continuous and closed interval

Question 14

Is there a number exactly one more than its cube?

Answer 14

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

Question 15

Do Limit Laws apply to infinite limits?

Answer 15

No, because infinity is not a number

Question 16

Whats infinity minus infinity?

Answer 16

DNE because infinity is not a number

Question 17

Answer 17

notice the method of multiplying by (1/x3) / (1/x3)

Ratio of leading coefficients.

Question 18

HARD problem

Answer 18

multiply by conjugates
Try to simplify getting numerator to 1
Can distribute multiplication into the square root.

Question 19

Tricky one.

Answer 19

USe squeeze theorem - think that when you see cos

Question 20

The equation of the derivative of a function f at a number a is….

Answer 20

limit as h goes to 0

Question 21

Write equation describing that the tangent line to y = f(x) at (a, f(x))

Answer 21

Y - f(a) = f’(a) (x - a)

Y changes as the rate of derivative times change in x

Question 22

Answer 22

write limit every time
Write the initial formula every time

Question 23

Whats the difference between average velocity and instantaneous velocity?

Answer 23

Average is given over two points

Instantaneous is a limit as h –> 0

Question 24

Are these all the same thing?

slope of tangent line
instantaneous rate of change
instantaneous velocity
marginal rate of change
derivative

Answer 24

yup

Question 25

T/F: If a function is differentiable, it is it continuous.

Answer 25

True. Differentiable means it has a derivative. And thus it is continuous.

The opposite is false though, it can be continuous but not differentiable

Question 26

Name some situations where a function is not differentiable

Answer 26

Discontinuous or has a corner

|x|
Has vertical asymptote
[[x]]

Question 27

How do you get the second derivative? And what does it mean?

Answer 27

Derivative of the first derivative.

It’s the rate of change of the rate of change. e.g. acceleration.

Question 28

What is third derivative called?

Answer 28

The jerk.

a large jerk means a sudden change in acceleration.

Question 29

Answer 29

If you see that subtraction at bottom, think of this formula - f(x) - f(a) / x-a

Question 30

Answer 30

Split into limit from left and right, piecewise

Question 31

What is difference of cubes simplification?

b3 - a3 = ?

Answer 31

(b-a)(b2 + ab + b2)

Question 32

For a function, how do you go about finding the horizontal and vertical asymptotes?

Answer 32

Horizonal: look at x –> infinity and negative infinity

Vertical: look at zeros of denominator. Then if you cannot factor that out, its a VA

Question 33

When have continuous function, the limit can pass through…

Answer 33

Question 34

What are various methods of calculating limits

Answer 34

Limit Laws
Direct substitution
Squeeze theorem
Conjugate method
Factoring
Limit translates inside lim f(g(x)) = f(lim g(x)) …. if f is continuous