Limits & Continuity
This term describes a point on a graph where a function's limit exists as x approaches a, but that limit does not equal f(a), often resembling a "hole" in the graph.
removable discontinuity
This term refers to the arbitrary constant that must be added to the result of every indefinite integral to represent all possible vertical shifts.
constant of integration
This feature of a graph occurs where the function's first derivative is zero and changes from positive to negative.
local (or relative) maximum
This term is defined as the absolute value of velocity, meaning it only measures magnitude and cannot be negative.
speed
This theorem states that if a function is continuous on a closed interval, it must take on every value between the outputs of its endpoints at least once.
Intermediate Value Theorem
This condition is met when the left-hand limit, right-hand limit, and the value of the function all equal each other at a given point.
continuity
This approximation method estimates the area under a curve by summing the areas of rectangles using the left endpoints, right endpoints, or midpoints of subintervals.
Riemann sum
If a function’s second derivative is strictly positive on an interval, the graph exhibits this geometric shape or orientation.
concave up
This quantity is found by taking the derivative of the velocity function with respect to time.
acceleration
This theorem guarantees that the instantaneous rate of change will equal the average rate of change at some point, provided the function is continuous and differentiable on the interval.
Mean Value Theorem
This specific type of asymptote occurs on a graph when the limit of a function as x approaches infinity or negative infinity is equal to a constant value, L.
horizontal asymptote
This integration technique reverses the Chain Rule by substituting a variable u for an inner function.
u-substitution
This term describes a straight line that a curve approaches arbitrarily closely as the coordinate moves toward infinity, but never quite touches.
asymptote
This term represents the total change in position of an object, calculated by evaluating the definite integral of velocity over a time interval.
displacement
This theorem allows you to evaluate a definite integral by subtracting the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit.
Fundamental Theorem of Calculus
This specific theorem guarantees that if a function is continuous on [a, b], it will attain both a maximum and a minimum value on that interval.
Extreme Value Theorem (EVT)
This term describes an integral with no upper and lower limits of integration, representing a family of functions rather than a specific numerical value.
indefinite integral
This specific point on a graph is found by locating where the second derivative is equal to zero or undefined, provided the concavity actually changes sign.
inflection point
To find this cumulative quantity, you must integrate the absolute value of the velocity function over a given time interval.
total distance traveled
This theorem states that a continuous function on a closed interval must achieve an absolute maximum value and an absolute minimum value at least once.
Extreme Value Theorem
This algebraic technique is used to evaluate limits that result in the indeterminate form 0/0 by taking the derivative of the numerator and denominator.
L'Hôpital's Rule
This method approximates the definite integral of a function by splitting the area into three-sided polygons with parallel vertical sides rather than rectangles.
Trapezoidal Rule
If a function has a derivative that is always negative on an interval, the graph of the function is doing this from left to right.
decreasing
A particle is said to be doing this when its velocity and acceleration have opposite signs.
slowing down
This theorem is a specific case of the Mean Value Theorem where the average rate of change between two endpoints happens to be exactly zero.
Rolle's Theorem