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Questions and Answers
What do we call a function F if its derivative is f?
What do we call a function F if its derivative is f?
What is the antiderivative of cos(x)?
What is the antiderivative of cos(x)?
Is the antiderivative of 3x^2 unique?
Is the antiderivative of 3x^2 unique?
How do we denote the antiderivative of a function f?
How do we denote the antiderivative of a function f?
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If F(x) is an antiderivative of f(x), what can F(x) + C represent?
If F(x) is an antiderivative of f(x), what can F(x) + C represent?
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What does the Fundamental Theorem of Calculus state about antiderivatives?
What does the Fundamental Theorem of Calculus state about antiderivatives?
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What is the area function of a constant function?
What is the area function of a constant function?
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What is the derivative of the area function of a constant function?
What is the derivative of the area function of a constant function?
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What type of function does a linear function's area function become?
What type of function does a linear function's area function become?
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On what interval is the area function of a linear function calculated in Example 2?
On what interval is the area function of a linear function calculated in Example 2?
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What happens to the area function of a linear function as 'b' increases?
What happens to the area function of a linear function as 'b' increases?
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What is the derivative of the area function of a linear function?
What is the derivative of the area function of a linear function?
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What is the derivative of $x^{a+1}$ with respect to $x$?
What is the derivative of $x^{a+1}$ with respect to $x$?
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What is the indefinite integral of $e^{ax}$ with respect to $x$?
What is the indefinite integral of $e^{ax}$ with respect to $x$?
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What does the area function $A(x)$ represent for a continuous function $f$?
What does the area function $A(x)$ represent for a continuous function $f$?
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What is the derivative of $ln(x)$ with respect to $x$?
What is the derivative of $ln(x)$ with respect to $x$?
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What is the indefinite integral of $sin(ax)$ with respect to $x$?
What is the indefinite integral of $sin(ax)$ with respect to $x$?
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Given the function $f(x) = x^2$, what would be the area under the curve from 0 to 2?
Given the function $f(x) = x^2$, what would be the area under the curve from 0 to 2?
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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.
