Define Accumulation Function.
A function defined by a definite integral where one of the limits of integration is a variable.
What does the Fundamental Theorem of Calculus relate?
Differentiation and integration; it shows they are inverse processes.
What is the integrand?
The function inside the integral that is being integrated.
What is a definite integral?
An integral with upper and lower limits, resulting in a numerical value.
What is an antiderivative?
A function whose derivative is the original function.
What is the upper bound of an integral?
The upper limit of integration in a definite integral.
What is the lower bound of an integral?
The lower limit of integration in a definite integral.
Define the Chain Rule.
A rule for differentiating composite functions.
What is the derivative?
The instantaneous rate of change of a function with respect to its variable.
What is integration?
The process of finding the area under a curve.
What is the formula for the Fundamental Theorem of Calculus (Part 1)?
$\frac{d}{dx}left[int_{a}^{x}f(t)dt
ight]=f(x)$
What is the formula for switching the bounds of integration?
$\int_{a}^{b}f(x) , dx = -\int_{b}^{a}f(x) , dx$
What is the formula for the Fundamental Theorem of Calculus (Part 1) with a function in the upper limit?
$\frac{d}{dx}left[int_{a}^{g(x)}f(t)dt
ight]=f(g(x)) cdot g'(x)$
What is the general form of an accumulation function?
$F(x) = \int_{a}^{x} f(t) , dt$
How to find $g'(x)$ if $g(x) = \int_{a}^{x} f(t) dt$?
Apply the Fundamental Theorem of Calculus: $g'(x) = f(x)$.
How to find $g'(x)$ if $g(x) = \int_{a}^{u(x)} f(t) dt$?
Apply the Fundamental Theorem of Calculus and the Chain Rule: $g'(x) = f(u(x)) cdot u'(x)$.
How to find $g'(c)$ given $g(x) = \int_{a}^{x} f(t) dt$?
First find $g'(x)$ using the Fundamental Theorem of Calculus, then evaluate $g'(c)$.
How to find $F'(x)$ if $F(x) = \int_{u(x)}^{b} f(t) dt$?
First, switch the limits: $F(x) = -\int_{b}^{u(x)} f(t) dt$. Then, $F'(x) = -f(u(x)) cdot u'(x)$.
How to find $g'(x)$ if $g(x) = \int_{u(x)}^{v(x)} f(t) dt$?
Split the integral: $g(x) = \int_{0}^{v(x)} f(t) dt - \int_{0}^{u(x)} f(t) dt$. Then, $g'(x) = f(v(x))v'(x) - f(u(x))u'(x)$.
How do you evaluate $\frac{d}{dx} \int_{2}^{x^3} \cos(t) dt$?
Apply the Fundamental Theorem of Calculus and the chain rule: $\cos(x^3) cdot 3x^2$.
How do you find $F'(x)$ if $F(x) = \int_{5}^{x} (t^2 + 1) dt$?
Apply the Fundamental Theorem of Calculus: $F'(x) = x^2 + 1$.
How do you solve for $g'(2)$ if $g(x) = \int_{1}^{x^2} \sqrt{t} dt$?
First find $g'(x) = \sqrt{x^2} cdot 2x = |x| cdot 2x$. Then, $g'(2) = \sqrt{2^2} cdot 2(2) = 8$.
How do you find the derivative of $\int_{x}^{0} e^{t^2} dt$?
Switch the bounds and apply the Fundamental Theorem of Calculus: $-\int_{0}^{x} e^{t^2} dt$, so the derivative is $-e^{x^2}$.
How do you find $h'(x)$ if $h(x) = \int_{\sin(x)}^{3} t^3 dt$?
Switch the bounds: $h(x) = -\int_{3}^{\sin(x)} t^3 dt$. Apply the Fundamental Theorem of Calculus and the chain rule: $h'(x) = -(\sin(x))^3 \cos(x)$.
