Integrals
This is the geometric meaning of a definite integral when the function lies entirely above the x-axis.
What is the area under the curve?
For a continuous, non-negative function on a closed interval [a, b], evaluating the definite integral from a to be yields this geometric measure of the region trapped between the curve and the x-axis.
What is area?
This fundamental calculus concept describes the value that a function approaches as the input gets arbitrarily close to a specific numbers, regardless of the function’s actual value at that point.
What is a limit?
This fundamental calculus concept simultaneously describes the instantaneous rate of change of a function and the slope of the line that just hits the curve at a single point
What is the derivative?
This rule allows you to evaluate limits that produce indeterminate forms such as 0/0 or infinity/infinity. By taking the limit of the ratio of the derivatives instead of the ratio of the original functions.
What is L’Hôpital’s Rule?
These sums estimate the exact value of a definite integral by partitioning an interval into sub intervals, creating rectangles whose heights depend on sample points like left endpoints, right endpoints, or midpoints, and adding the rectangular areas together.
What are Riemann Sums?
This quantity represents the net change in a particle’s position over a time interval and is found by integrating the velocity function directly so that intervals of negative velocity cancel intervals of positive velocity to find a final answer.
What is displacement.
For a two-sided limit to exist at a point, these two one-sided limits must not only exist but also be identical
What are the left-hand and right-hand limits?
When taking the derivative of a composition like sin(x^2) or e^3x, you apply this rule, which instructs you to differentiate the outside of the function, leave the inside alone, then multiple by the derivative of the inside.
What is the Chain Rule?
When analyzing the motion of a particle along a staring line, the particle is slowing down whenever these two quantities have opposite signs, and speeding up when they share the same sign.
What are Velocity and Acceleration?
Part 2 of this key theorem provides the primary method for evaluating definite integrals exactly, stating that the integral from a-b of f(x)do is equal to F(b) - F(a). Where F is any antiderivative of f
What is the Fundamental Theorem of Calculus?
When computing the area between two curves, this essential preliminary step determines both the limits of integration and reveals which function serves as the upper boundary on each subinterval.
What is locating the points of intersection?
A function has this type of discontinuity at x=c when the limit as x approaches c exists, but the function is either undefined at c or its value there does not equal the limit
What is a removable discontinuity?
When finding the derivative of a number with difficult numbers/other numbers you use this rule where you find the derivative of the top number, times the bottom, minus the top number, times the derivative of the bottom number, all over the bottom number squared.
What is the Quotient rule?
These are the only possible x-values where a function can have a local maximum or minimum; they occur where the derivative equals zero or faults to exist.
What are critical numbers/points?
Unlike a normal integral, which values to a single number, this type of integral is written without bounds and represents the complete family of anti derivatives of a function, always including an arbitrary constant of integration.
What is an indefinite integral?
When a region bounded by two curves is revolved around an axis and the inner curve creates a hollow space through the solid, this volume method subtracts the inner disk’s volume from the outer disk’s volume.
What is the washer method?
This theorem states that if a function is continuous on a closed interval [a, b], then it must take on every value between f(a) and f(b) at least once within that interval.
What is the Intermediate Value Theorem, or IVT?
Geometrically, this sign of this higher-order derivative determines whether a curve is concave up or concave down.
What is the second derivative?
This theorem guarantees that if a function is continuous on a closed interval [a, b] And differentiable on the open interval (a, b), then tehre exists at least one number c in (a, b) where the instantaneous rate of change equals the average rate of change over [a, b]
What is the Mean Value Theorem?
The integral of arctangent(X).
What is 1/2ln(1+x^2)?
This quantity, defined as 1/b-a integral of a-b f(x)do and frequently confused with the average rate of change - geometrically represents the height of a rectangle with base (b-a) whose area equals the net signed area between the curve and the x-axis over the interval.
What is the average value of a function?
This type of discontinuity occurs when a function oscillates infinitely between two values as x approaches c, meaning neither the left-hand nor the right-hand limit exists due to the functions nature.
What is an oscillating discontinuity?
In kinematics, the first derivative of position yields velocity and the second yields acceleration; what derivative measures the rate at which acceleration itself changes over time.
What is the third derivative?
This theorem guarantees that a function continuous on a closed interval [a, b] must attain an absolute maximum value and an absolute minimum value somewhere on that interval, though it does not specify where those extrema actually occur.
What is the Extreme Value Theorem?