Basic Calculus - Midterm Quiz (Retake)

Which of the following describes a jump discontinuity?

• The function is undefined at a certain point.

• The function approaches positive or negative infinity as it approaches a point.

• The function oscillates infinitely at a point.

• The function approaches different finite values from the left and the right at a point.

At which type of discontinuity does the limit of the function exist, but the function is not defined at that point?

• Removable discontinuity

• Jump discontinuity

• Infinite discontinuity

• Essential discontinuity

Let h be a continuous function on the closed interval [-3,4],

h(-3)=-1 and h(4)=2. Which of the following is guaranteed by the Intermediate Value Theorem?

• h(c)=-2 for at least one c between-1 and 2

• h(c)=1 for at least one c between-3 and 4

• h(c)=-2 for at least one c between-3 and 4

• h(c)=1 for at least one c between-1 and 2

Let g(x)=x^3-12x+7. The absolute maximum value of g over the closed interval [-4,5] occurs at what value x-value?

• 4

• -4

• 2

• 5

Consider f(x)=3x^3+8. What is the absolute minimum value of f over the closed interval -2≤x≤2?

• -16

• 16

• 8

• -8

Consider a function f(x) which is continuous on the interval [a, b]. Which of the following statements regarding the Intermediate Value Theorem is true?

• The function must attain every value between f(a) and f(b) at exactly one point in the interval (a, b).

• The function must attain every value between f(a) and f(b) at the points and b.

• The Intermediate Value Theorem only applies to polynomials.

• The function must attain every value between f(a) and f(b) at some point in the interval (a, b).

Suppose f(x) is a continuous function on a closed interval [a, b]. Which of the following statements about the Extreme Value Theorem is correct?

• The function must have a local maximum and a local minimum somewhere in the interval (a, b).

• The function must have either an absolute maximum or an absolute minimum somewhere in the interval [a, b].

• The function must have both an absolute maximum and an absolute minimum somewhere in the interval [a, b].

• The Extreme Value Theorem only applies to differentiable functions.

Suppose f(x) is a continuous function on the closed interval [0, 1] such that f(0)=2 and f(1)=5. Which of the following statements is guaranteed by the Intermediate Value Theorem and the Extreme Value Theorem?

• There exists a point c in the interval (0, 1) such that f(c)=3.

• There exists a point c in the interval [0, 1] such that f(c)=6.

• There exists a point c in the interval [0, 1] such that f(c)=2 or f(c)=5.

• There exists a point c in the interval (0, 1) such that f(c)=2 or f(c)=5.

Suppose f(x) is a continuous function on the interval [0,5], and f(0)=4 and

f(5)=12. Which of the following statements is true?

• There exists a value [0,5] such that f(c)=0.

• There exists a value [0,5] such that f(c)=2.

• There exists a value [0,5] such that f(c)=10.

• There exists a value [0,5] such that f(c)=14.

Consider the following scenario involving a continuous function f(x) defined on the closed interval [a,b]:

A cyclist is riding along a straight path, and her position at each moment is described by a continuous function f(x) where x represents time in minutes, and f(x) represents the distance covered in kilometers from the starting point. The cyclist starts her journey at time x=0 and finishes at time x=10, covering a total distance of 50 kilometers.

Which theorem guarantees that at some point during her journey, the cyclist must have been exactly 25 kilometers away from the starting point AND the cyclist reaches her furthest distance from the starting point at some point during her journey?

• Both Intermediate Value Theorem

• Both Extreme Value Theorem

• Intermediate Value Theorem, Extreme Value Theorem

• Extreme Value Theorem, Intermediate Value Theorem