Test 2

Question 1

State the Extreme Value Theorem carefully

Answer 1

If f is continuous on a closed interval [a, b], then we can find M and m in [a, b] such that:
f(M) is the maximum value of f on [a, b]
f(m) is the minimum value of f on [a, b]

Question 2

Why does the Extreme Value Theorem require closed intervals?

Answer 2

Because it cannot guarantee global extrema, but it can guarantee a local minimum and maximum, which is why it must have closed intervals.

Question 3

What is right continuity?

Answer 3

It means that a limit exists and when plugged back in it equals f(x), plus it’s a positive exponent so h to the 2 positive on top

Question 4

What is left continuity?

Answer 4

It means that a limit exists and when plugged in it will equal f(x), but this time it is a negative exponent. So h to the 2 negative exponent.

Question 5

f is continuous on the open interval (a, b), what does this mean?

Answer 5

A function is continuous over an open interval if it is continuous at every point in the interval

Question 6

f is continuous on the closed interval [c, d], what does this mean?

Answer 6

It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints

Question 7

What is the difference between lim[3] f(x) and f(3)?

Answer 7

lim[3] f(x) = means that the values are arbitrarily close to that value (3)

f(3) = a calculation has been done at a certain point

Question 8

What is the difference between lim → 2+ f(x), lim → 2- f(x), and lim → 2 f(x)?

Answer 8

One is coming from the left (smaller than 2) one is coming from the right (bigger than 2) and one is free flowing from all directions.

Question 9

Why are polynomials continuous functions?

Answer 9

A polynomial function is a sum of powers of x and since powers of x are continuous functions then every polynomial function is therefore continuous.

Question 10

What is a secant line?

Answer 10

Straight line joining two points at a function

Question 11

What is a tangent line?

Answer 11

A straight line touching one point

Question 12

What is the difference between the average rate of change of a function and the instantaneous rate of change of a function?

Answer 12

The average rate of change calculates the slope of the secant line using the slope formula from algebra.

The instantaneous rate of change calculates the slope of the tangent line using derivatives.

Question 13

What is the equation for the horizontal asymptote test?

Answer 13

b x to the n power

Question 14

HA test results…

Answer 14

m < n = HA = 0
m > n = HA = HA does not exist
m = n = HA = a/b

Question 15

What is the hypothesis of the Extreme Value Theorem?

Answer 15

f be continuous on a closed interval [a, b] is essential.

Question 16

What is the conclusion of the Extreme Value Theorem?

Answer 16

If f either fails to be continuous on the interior of the interval or fails to be continuous at a closed endpoint, then the conclusion of these theorems do not necessarily hold.

Question 17

State the constant multiple law

Answer 17

lim k f(x) = k(lim f(x))

Question 18

State the sum law

Answer 18

lim [f(x) + g(x)] = (lim f(x)) + (lim g(x))

Question 19

State the product law

Answer 19

lim[f(x) times g(x)] = (lim f(x)) times (lim g(x))

Question 20

State the quotient law

Answer 20

lim f(x)/g(x) = (lim f(x)) / (lim g(x))

Question 21

how is the derivative of a function at point x = c related to the average rate of change

Answer 21

The point is on a secant line

Question 22

how is the derivative of a function at point x = c related to the instantaneous rate of change

Answer 22

The point will be on a tangent line

Question 23

When drawing the derivative from the original function, what are the rules?

Answer 23

max/mins = x-int
decreasing = under x
increasing = over x
slope = y-int

Question 24

Name a function that is continuous, but not differential at a point

Answer 24

f(x) = |x| on point (0, 0)

Question 25

What is an antiderivative F of a function f?

Answer 25

x squared + C

Question 26

Give the technical definition of “f is continuous at x = 1”

Answer 26

f is continuous at 1 = lim x tends to 1 f(x) = f(1)

Question 27

True or false: When applying the rules (sum, product, etc.) you need to repeat the limit before plugging it in on every number?

Answer 27

True

Question 28

What is the equation of finding a derivative with x and h

Answer 28

lim tends to 0 = f(x+h) - f(x) over h

Question 29

What is the derivative equation with h

Answer 29

lim tends to c f(c+h) - f(c) / h

Question 30

Name the three types of discontinuities

Answer 30

Jump, removable, and infinite