Limits & Continuity
This value describes what a function approaches as x approaches a number.
What is a limit?
This is the instantaneous rate of change of a function.
What is the derivative?
The integral of 1/x dx.
What is ln|x| + C?
Find the derivative of f(x)=x3−4x.
What is 3x2−4?
This theorem states that if a function is continuous on [a, b], it has at least one point where the instantaneous rate equals the average rate.
What is the Mean Value Theorem?
This type of discontinuity occurs when a function approaches infinity.
What is an infinite discontinuity?
The derivative of sin(x).
What is cos(x)?
This is the reverse process of differentiation.
What is integration?
Evaluate ∫(3x2−2)dx.
What is x3−2x+C?
This rule is used to differentiate compositions of functions.
What is the chain rule?
Evaluate:
lim
x→3 (2x+1).
What is 7?
The derivative of ex is this.
What is ex?
This method uses substitution to simplify integrals.
What is u-substitution?
Find the derivative of f(x)=5x4.
What is 20x3?
These are small line segments that represent slopes at points in the plane.
What is a slope field?
This type of discontinuity occurs when a function has a “hole” that could be fixed by redefining the function’s value at that point.
What is a removable discontinuity?
This is what the derivative represents geometrically at a point on a curve.
What is the slope of the tangent line?
This technique is used for integrating products of functions and is based on the product rule.
What is integration by parts?
Evaluate ∫xex dx.
What is xex−ex+C?
This numerical method approximates solutions to differential equations using tangent line steps.
What is Euler's Method?
Evaluate:
lim
x→0 sinx/x.
What is 1?
If the derivative changes from positive to negative, the function has this at that point.
What is a local maximum?
This type of integral has infinite limits or discontinuities.
What is an improper integral?
Evaluate ∫1/(1+x2) dx.
What is arctan(x)+C?
The derivative of position with respect to time gives this quantity.
What is velocity?