Ch2 Limits and Derivatives

Question 1

Constant Multiple Law (Limits)

Answer 1

The limit is concerned with what value the function is approaching as
𝑥 →𝑎. Multiplying the whole function by a fixed number just stretches or compresses that value — it doesn’t interfere with the approach itself.

Question 2

Squeeze Theorem

Answer 2

If a function is trapped between two others, and those two are heading to the same limit, then the trapped function must go there too.

Question 3

Vertical Asymptotes

Answer 3

Occur when a function is undefined AND, at that point, the functions value balloons to infinity

Question 4

Horizontal Asymptotes

Answer 4

Occur when, as x approaches infinity, y approaches a certain value

Question 5

Intermediate Value Theorem

Answer 5

If a function is continuous (no jumps, holes, or breaks), and it goes from one value to another over an interval, then it must pass through every y-value in between.

Question 6

Derivative Formal Definition

Answer 6

A derivative f’(a) is the slope of the tangent line to the graph f(x) at x=a AND the instantaneous rate of change of f at x=a

Question 7

First Derivative

Answer 7

The first derivative gives you the rate of change of your functions position (your speed/velocity).

Question 8

Second Derivative

Answer 8

The second derivative gives you the rate of change of your speed (how fast your speed is changing). This is called your acceleration

Question 9

Differentiability

Answer 9

A function is differentiable at a point if the slope is the same on either side of that point; the slope is smooth - you can find it. No sudden turns.

Question 10

Differentiability vs. Continuity

Answer 10

If a function is differentiable at a, it is continuous at a. A function can be continuous at a, but not differentiable at a (it could have a sharp corner or cusp)

Question 11

Continuity

Answer 11

You can draw the whole function without lifting your hand. No jumps or holes.

Question 12

Finding the Avg. Velocity in a Time Period

Answer 12

Question 13

Instantaneous Velocity

Answer 13

Derivative of position

Question 14

When f(x) is increasing, f’(x):

Answer 14

f’(x) > 0

Question 15

When f(x) is decreasing, f’(x):

Answer 15

f’(x) < 0

Question 16

When f(x) has a peak or valley (max or min), f’(x):

Answer 16

f’(x) = 0

Question 17

When f(x) is increasing or decreasing STEEPLY, f’(x):

Answer 17

has a large y value (is far above or below the x axis)

Question 18

When f(x) has sharp corners or cusps, f’(x):

Answer 18

is undefined

Question 19

For rational functions, if the degree of the numerator < the degree of the denominator,

Answer 19

there is a horizontal asymptote at y = 0

Question 20

For rational functions, if the degree of the numerator is > the degree of the denominator,

Answer 20

there is no horizontal asymptote, but there COULD be a slant asymptote

Question 21

For rational functions, if the degree of the numerator is = to the degree of the denominator,

Answer 21

there is a horizontal asymptote at y = (the leading coefficient of the numerator / the leading coefficient of the denominator)

Question 22

What are the 7 steps to sketching a curve?

Answer 22

1. Domain
2. X and Y intercepts
3. Symmetry (even, odd, periodic)
4. Asymptotes
5. Intervals of Increase/Decrease
6. Maxima and Minima
7. Concavity and points of inflection

Question 23

Antiderivative of

Answer 23

where n cannot = -1

Question 24

Antiderivative of

Answer 24

x cannot = 0

Question 25

Antiderivative of

Answer 25

Question 26

Antiderivative of

Answer 26

a > 0, a cannot = 1

Question 27

Antiderivative of

Answer 27

Question 28

Antiderivative of

Answer 28