6.1 Antiderivatives Graphically and Numerically

Question 1

What is an antiderivative of a function say f(x)?

Answer 1

An antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) is f(x), or F′(x)=f(x).

Question 2

How is the antiderivative F(x) related to the area under a curve?

Answer 2

The antiderivative F(x) represents the accumulated area under the curve f(x) from a starting point.

Question 2

What is the difference between an antiderivative and a definite integral?

Answer 2

The antiderivative is a general function that includes a constant C, while the definite integral computes the total accumulated area over a specific interval.

Question 3

How can you estimate an antiderivative numerically?

Answer 3

using methods like Riemann sums, which approximate the area under the curve by summing up rectangles.

Question 4

What is the relationship between the f(x) graph and its antiderivative F(x) in terms of slope?

Answer 4

The slope of the graph of the antiderivative F(x) at any point is the value of the function f(x) at that point, i.e., F′(x)=f(x).

Question 5

How do you graph an antiderivative?

Answer 5

The antiderivative is a curve whose slope depends on the function. If
f(x) is positive, the curve goes up; if f(x) is negative, the curve goes down. The steeper f(x) is, the faster the antiderivative increases or decreases.

Question 5

What is the constant of integration, what does it represent, when is it added and why?

Answer 5

The constant of integration,
C, represents an unknown constant that is added when finding an indefinite antiderivative since the differentiation of a constant is zero.