18 Questions
What do we call a function F if its derivative is f?
Antiderivative of f
What is the antiderivative of cos(x)?
Sin(x)
Is the antiderivative of 3x^2 unique?
No, it has multiple possibilities
How do we denote the antiderivative of a function f?
Indefinite integral
If F(x) is an antiderivative of f(x), what can F(x) + C represent?
A different function
What does the Fundamental Theorem of Calculus state about antiderivatives?
The derivative of an antiderivative is the original function
What is the area function of a constant function?
Constant
What is the derivative of the area function of a constant function?
Zero
What type of function does a linear function's area function become?
Linear
On what interval is the area function of a linear function calculated in Example 2?
[0, b]
What happens to the area function of a linear function as 'b' increases?
It always increases
What is the derivative of the area function of a linear function?
A constant value
What is the derivative of $x^{a+1}$ with respect to $x$?
$(a+1)x^a$
What is the indefinite integral of $e^{ax}$ with respect to $x$?
$ae^{ax} + C$
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 $t = x$
What is the derivative of $ln(x)$ with respect to $x$?
$rac{1}{x}$
What is the indefinite integral of $sin(ax)$ with respect to $x$?
$-cos(ax) + C$
Given the function $f(x) = x^2$, what would be the area under the curve from 0 to 2?
5.0
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.
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.
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