Ch2 Limits and Derivatives

Question 1

Sum Law (Limits)

Answer 1

Question 2

Constant Multiple Law (Limits)

Answer 2

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 3

Power Law (Limits)

Answer 3

Question 4

Root Law (Limits)

Answer 4

Question 5

Product Law (Limits)

Answer 5

Question 6

Quotient Law (Limits)

Answer 6

Question 7

Squeeze Theorem

Answer 7

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 8

Vertical Asymptotes

Answer 8

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

Question 9

Horizontal Asymptotes

Answer 9

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

Question 10

Intermediate Value Theorem

Answer 10

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 11

Derivative Formal Definition

Answer 11

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 12

First Derivative

Answer 12

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

Question 13

Second Derivative

Answer 13

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

Question 14

Differentiability

Answer 14

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 15

Differentiability vs. Continuity

Answer 15

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 16

Continuity

Answer 16

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

Question 17

Finding the Avg. Velocity in a Time Period

Answer 17

Question 18

Instantaneous Velocity

Answer 18

Derivative of position

Question 19

When f(x) is increasing, f’(x):

Answer 19

f’(x) > 0

Question 20

When f(x) is decreasing, f’(x):

Answer 20

f’(x) < 0

Question 21

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

Answer 21

f’(x) = 0

Question 22

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

Answer 22

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

Question 23

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

Answer 23

is undefined