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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?
- Antiderivative of f (correct)
- Definite integral of f
- Derivative of f
- Integral of f
What is the antiderivative of cos(x)?
What is the antiderivative of cos(x)?
- Csc(x)
- Tan(x)
- Sec(x)
- Sin(x) (correct)
Is the antiderivative of 3x^2 unique?
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?
How do we denote the antiderivative of a function f?
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?
What does the Fundamental Theorem of Calculus state about antiderivatives?
What does the Fundamental Theorem of Calculus state about antiderivatives?
What is the area function of a constant function?
What is the area function of a constant function?
What is the derivative of the area function of a constant function?
What is the derivative of the area function of a constant function?
What type of function does a linear function's area function become?
What type of function does a linear function's area function become?
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?
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?
What is the derivative of the area function of a linear function?
What is the derivative of the area function of a linear function?
What is the derivative of $x^{a+1}$ with respect to $x$?
What is the derivative of $x^{a+1}$ with respect to $x$?
What is the indefinite integral of $e^{ax}$ with respect to $x$?
What is the indefinite integral of $e^{ax}$ with respect to $x$?
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 $f$?
What is the derivative of $ln(x)$ with respect to $x$?
What is the derivative of $ln(x)$ with respect to $x$?
What is the indefinite integral of $sin(ax)$ with respect to $x$?
What is the indefinite integral of $sin(ax)$ with respect to $x$?
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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