AMETRINE 24-25

• The limit of the function as x approaches c

• The value of the function at x = c

• The slope of the function at x = c

• The derivative of the function at x = c

Answer explanation

The notation f(c) represents the value of the function at x=c, meaning the output of the function when the input is exactly c.

Which of the following statements is TRUE about the limit of a function?

• The limit is always equal to the function value at that point.

• The limit is the highest possible value of the function.

• The limit is the lowest possible value of the function.

• The limit is the value the function approaches as the input approaches a certain value

Answer explanation

The limit of a function at a certain point is the value that the function approaches as the input gets arbitrarily close to that point, regardless of whether the function is actually defined there. This means that even if the function has a hole or a discontinuity at that point, the limit can still exist and be different from the actual function value.

What is the purpose of illustrating the limit of a function using a table of values and the graph of the function?

• To show the maximum value of the function.

• To show the minimum value of the function.

• To show how the function behaves near a specific point.

• To show the slope of the function at a specific point.

Answer explanation

The limit of a function shows what number the function is getting closer to as we choose values closer and closer to a certain point.

Which of the following indicates an open interval?

• (a,b]

• [a,b)

• (a,b)

• [a,b]

Answer explanation

An open interval is denoted by parentheses ( ).

Which of the following statements is true regarding the continuity of a function f(x) on the closed interval [a, b]?

• f(x) is continuous at every point in the interval.

• f(x) is continuous at every point in the interval except possibly at a and b.

• f(x) is continuous at a and b, but not necessarily at any other point in the interval.

• f(x) is continuous at every point in the interval except possibly at finitely many points.

Answer explanation

In the closed interval [a,b] the function is said to be continuous to the points within the interval including a and b.

What is the condition for a function to be continuous at a point?

• The limit at the point exists.

• The function is defined at the point

• The function is differentiable at the point

• The limit at the point does not exist

Answer explanation

One essential condition for a function to be continuous is that the function must have a limit.

What line touches a curve at one point?

• Derivative

• Normal

• Secant

• Tangent

Answer explanation

Tangent line is a line that touches the curve at exactly one point.

If f(x) = 2x - 1, what is f(x + ∆x) - f(x)?

• 2(x + ∆x) - 1

• 2(x + ∆x) - 1 - (2x - 1)

• (2x + ∆x) - 1 - (2x - 1)

• 2(x + ∆x) - 1 - 2(x - 1)

Answer explanation

To find f(x+Δx)−f(x), substitute x+Δx into the function:

f(x+Δx) = 2(x+Δx)−1

f(x) = 2x -1

Then subtract f(x):

f(x+Δx) − f(x) = 2(x+Δx) − 1 − (2x−1).

What is the rate of change(derivative) of a function f(x) at x = a?

• It is equal to f(a).

• It is the limit of f(x) as x approaches a.

• It is equal to the slope of the tangent at x = a

• It is equal to the average rate of change for an interval of x containing a.

Answer explanation

rate of change means slope of the tangent line which is also equal to the derivative

What tells the rate a moving object is changing its velocity at a certain point in time?

• acceleration

• average velocity

• instantaneous velocity

• terminal velocity

Answer explanation

Acceleration tells us how quickly an object's velocity is changing over time. It describes the rate of change of velocity at a particular moment.