Antiderivative Quiz: Practice Indefinite Integrals with Mathematics 105

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Questions and Answers

What do we call a function F if its derivative is f?

Antiderivative of f (correct)

Definite integral of f

Derivative of f

Integral of f

What is the antiderivative of cos(x)?

Csc(x)

Tan(x)

Sec(x)

Sin(x) (correct)

Is the antiderivative of 3x^2 unique?

Yes, but with a specific condition

No, it has multiple possibilities (correct)

Yes, always unique

No, it depends on x

How do we denote the antiderivative of a function f?

Indefinite integral Signup and view all the answers

If F(x) is an antiderivative of f(x), what can F(x) + C represent?

A different function Signup and view all the answers

What does the Fundamental Theorem of Calculus state about antiderivatives?

The derivative of an antiderivative is the original function Signup and view all the answers

What is the area function of a constant function?

Constant Signup and view all the answers

What is the derivative of the area function of a constant function?

Zero Signup and view all the answers

What type of function does a linear function's area function become?

Linear Signup and view all the answers

On what interval is the area function of a linear function calculated in Example 2?

[0, b] Signup and view all the answers

What happens to the area function of a linear function as 'b' increases?

It always increases Signup and view all the answers

What is the derivative of the area function of a linear function?

A constant value Signup and view all the answers

What is the derivative of $x^{a+1}$ with respect to $x$?

$(a+1)x^a$ Signup and view all the answers

What is the indefinite integral of $e^{ax}$ with respect to $x$?

$ae^{ax} + C$ Signup and view all the answers

What does the area function $A(x)$ represent for a continuous function $f$?

The signed area under the curve $y = f(t)$ between $t = a$ and $t = x$ Signup and view all the answers

What is the derivative of $ln(x)$ with respect to $x$?

$rac{1}{x}$ Signup and view all the answers

What is the indefinite integral of $sin(ax)$ with respect to $x$?

$-cos(ax) + C$ Signup and view all the answers

Given the function $f(x) = x^2$, what would be the area under the curve from 0 to 2?

5.0 Signup and view all the answers

Study Notes

Computation of Area Function

The area function of a constant function is linear.
The area function of a linear function is quadratic.

Antiderivatives or Indefinite Integrals

If F and f are two functions on [a, b] such that F'(x) = f(x), then f is the derivative of F, and F is an antiderivative of f.
An antiderivative of f is denoted by an indefinite integral: ∫f(x) dx = F(x) + C.
The derivative of an antiderivative F is unique, but the antiderivative of f is not unique, but determined up to an additive constant.

Examples of Indefinite Integrals

Power rule: ∫x^a dx = x^(a+1)/(a+1) + C if a ≠ -1, and ln|x| + C if a = -1.
Exponential and trigonometric functions:
- ∫e^(ax) dx = (1/a)e^(ax) + C
- ∫sin(ax) dx = (-1/a)cos(ax) + C
- ∫cos(ax) dx = (1/a)sin(ax) + C
- ∫sec^2(ax) dx = (1/a)tan(ax) + C
- ∫sec(ax)tan(ax) dx = sec(ax) + C

Definite vs Indefinite Integral

The area function A(x) of a continuous function f on [a, b] is defined as the definite integral: A(x) = ∫[a, x] f(t) dt.
A(x) is a function on [a, b] whose value at x is the signed area under the curve y = f(t) between t = a and t = x.

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Description

Test your understanding of antiderivatives and indefinite integrals in mathematics. This quiz covers the definitions of antiderivatives, derivatives, definite and indefinite integrals, and the Fundamental Theorem of Calculus. Perfect for students studying Math 105 (Section 204) or anyone interested in calculus concepts.