8.1.2 Three Big Theorems

Question 1

Three Big Theorems

Answer 1

• The intermediate value theorem: If f is continuous on a closed interval [a, b] and Q is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = Q.
• Rolle’s theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is at least one number c in (a, b) such that f (c) = 0.
• The mean value theorem: If f is continuous on [a, b] and differentiable on (a, b), then there is at least one number c in (a, b) such that f(b) - f(a) / b - a = f’(c)

Question 2

note

Answer 2

• The intermediate value theorem proves that continuous functions cannot jump around. If a continuous function goes through two different points, it must also go through each and every value between the two points.
• Rolle’s theorem is about tangent lines. Specifically, Rolle’s theorem says that if a continuous, differentiable function produces the same range value for two different domain values, then there must be a point between the two domain values where the function has a horizontal tangent.
• The mean value theorem is a generalization of Rolle’s
  theorem. Instead of looking for horizontal tangents, the mean value theorem says that there will be at least one tangent line with slope equal to the slope of the secant line connecting the two specified points.

Question 3

Given f (1) = 3 and f (5) = 11. If f is continuous on [1, 5], the intermediate value theorem guarantees which of the following statements about f for some c in the interval (1, 5).

Answer 3

f (c) = 9

Question 4

Given f (1) = −3 and f (5) = 9. If f is continuous on [1, 5] and differentiable on (1, 5), the mean value theorem guarantees which of the following statements about f for some c in the interval (1, 5).

Answer 4

f ′(c) = 3

Question 5

Which of the following theorems states that a function continuous on a closed interval takes on all intermediate range values?

Answer 5

The intermediate value theorem

Question 6

Suppose you have a continuous and differentiable function on a closed interval from a to b. Which of the following statements must be true?

Answer 6

There is at least one point c such that the slope of the line tangent to the function at c is equal to the slope of the secant line through the function at a and b

Question 7

Given f (1) = 3 and f (5) = 11. If f is continuous on [1, 5] and differentiable on (1, 5), the mean value theorem guarantees which of the following statements about f for some c in the interval (1, 5)

Answer 7

f ′(c) = 2

Question 8

Given f (3) = 7 and f (5) = 7. If f is continuous on [3, 5] and differentiable on (3, 5), Rolle’s theorem guarantees which of the following statements about f for some c in the interval (3, 5).

Answer 8

f ′(c) = 0

Question 9

Given f (1) = −3 and f (5) = 9. If f is continuous on [1, 5], the intermediate value theorem guarantees which of the following statements about f for some c in the interval (1, 5).

Answer 9

f (c) = 0

Question 10

Suppose you are given the parabola y = x^ 2. A point must exist on the graph of the function where the line tangent to the curve is horizontal. This observation is a consequence of which of the following theorems?

Answer 10

Rolle’s theorem

Question 11

Given f (−3) = −1 and f (3) = −1. If f is continuous on [−3, 3] and differentiable on (−3, 3), Rolle’s theorem guarantees which of the following statements about f for some c in the interval (−3, 3)

Answer 11

f ′(c) = 0

Question 12

Suppose a function is continuous on the closed interval from a to b and is differentiable on the open interval from a to b everywhere except a single point. That there must be a point c where the slope of the line tangent to the function is equal to the slope of the secant line passing through a and b is a consequence of which theorem?

Answer 12

No theorem justifies this statement.