zuai-logo

What is the Fundamental Theorem of Calculus (Part 1)?

ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t) dt = f(x)

All Flashcards

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?
$$lw$$
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 are the differences between a critical point and a point of inflection?
Critical Point: $f'(x) = 0$ or undefined, potential for max/min. | Point of Inflection: $f''(x)$ changes sign, change in concavity.
What are the differences between relative and absolute extrema?
Relative Extrema: Local max/min within an interval. | Absolute Extrema: Overall max/min over the entire domain.
What are the differences between increasing and concave up?
Increasing: $f'(x) > 0$, function is rising. | Concave Up: $f''(x) > 0$, function curves upwards.
What are the differences between decreasing and concave down?
Decreasing: $f'(x) < 0$, function is falling. | Concave Down: $f''(x) < 0$, function curves downwards.
What are the differences between the graph of a function and its derivative?
Function: Represents the value of the function at each point. | Derivative: Represents the rate of change of the function at each point.
What is the difference between a definite and an indefinite integral?
Definite Integral: Computes the area under a curve between two limits, resulting in a numerical value. | Indefinite Integral: Finds the antiderivative of a function, resulting in a family of functions.
What is the difference between $f'(x)$ and $\int f(x) dx$?
$f'(x)$: The derivative of $f(x)$, representing the instantaneous rate of change. | $\int f(x) dx$: The antiderivative of $f(x)$, representing the accumulation of $f(x)$.
What is the difference between using the first derivative test and the second derivative test to find relative extrema?
First Derivative Test: Examines the sign change of $f'(x)$ around a critical point. | Second Derivative Test: Uses the sign of $f''(x)$ at a critical point to determine concavity and thus whether it is a max or min.
What is the difference between a local extremum and an endpoint extremum?
Local Extremum: A maximum or minimum within the interior of an interval. | Endpoint Extremum: A maximum or minimum that occurs at the boundary of an interval.
What is the difference between average rate of change and instantaneous rate of change?
Average Rate of Change: The slope of the secant line between two points. | Instantaneous Rate of Change: The slope of the tangent line at a single point, given by the derivative.