Revise later
SpaceTo flip
If confident
All Flashcards
Define Accumulation Function.
A function that represents the accumulated area under a curve from a fixed point to a variable point.
What is an antiderivative?
A function whose derivative is the given function.
Define the Fundamental Theorem of Calculus.
The theorem that links the concept of the integral of a function with the concept of the derivative of a function.
What is a point of inflection?
A point on a curve where the concavity changes.
Define relative maximum.
A point where the function's value is greater than or equal to the values at all nearby points.
Define relative minimum.
A point where the function's value is less than or equal to the values at all nearby points.
Define Concave Up.
A curve that opens upwards.
Define Concave Down.
A curve that opens downwards.
What is the area under the curve?
The integral of a function between two points, representing the accumulation of the function's values.
Define critical point.
A point where the derivative of a function is either zero or undefined.
What does the Fundamental Theorem of Calculus (Part 1) state?
If $f$ is continuous on $[a, b]$, then the function $F$ defined by $F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $F'(x) = f(x)$.
What does the Fundamental Theorem of Calculus (Part 2) state?
If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$.
How does the Fundamental Theorem of Calculus relate differentiation and integration?
It shows that differentiation and integration are inverse processes.
What is the Mean Value Theorem for Integrals?
If $f$ is continuous on $[a, b]$, then there exists a number $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$.
What is the Intermediate Value Theorem?
If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $(a, b)$ such that $f(c) = k$.
How can the Fundamental Theorem of Calculus be used to find the area under a curve?
By finding the antiderivative of the function and evaluating it at the limits of integration.
What is the relationship between the derivative of an integral and the original function according to the Fundamental Theorem of Calculus?
The derivative of the integral of a function is the original function itself.
What is the Squeeze Theorem?
If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$.
What does the Extreme Value Theorem state?
If $f$ is continuous on a closed interval $[a, b]$, then $f$ has both a maximum and a minimum value on that interval.
What is Rolle's Theorem?
If a function $f$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval $(a, b)$ such that $f'(c) = 0$.
What is the Fundamental Theorem of Calculus (Part 1)?
$$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$
What is the formula for area under a curve?
$$\int_a^b f(x) dx$$
If $F(x) = \int_a^x f(t) dt$, what is $F'(x)$?
$$F'(x) = f(x)$$
If $F(x) = \int_a^x f(t) dt$, what is $F''(x)$?
$$F''(x) = f'(x)$$
How to find $f(0)$ given $f(4)$ and $f'(x)$?
$$f(0) = f(4) - \int_0^4 f'(x) dx$$
How to find $f(5)$ given $f(4)$ and $f'(x)$?
$$f(5) = f(4) + \int_4^5 f'(x) dx$$
Formula for $g'(x)$ if $g(x) = f(x) - x$?
$$g'(x) = f'(x) - 1$$
Area of a semicircle?
$$\frac{\pi}{2}r^2$$
Area of a triangle?
$$\frac{1}{2}bh$$
Area of a rectangle?
$$lw$$