Limits & Continuity
What does the limit of a function represent?
The limit represents the value a function approaches as the input approaches a certain point.
What does the derivative of a function represent?
The derivative represents the instantaneous rate of change of a function, or the slope of the tangent line at a point.
What is the purpose of a definite integral?
A definite integral finds the net area under a curve between two bounds, representing accumulated change.
What is an infinite series?
An infinite series is the sum of the terms of an infinite sequence.
What are parametric equations?
Parametric equations express both x and y as functions of a third variable, typically t, to describe motion or curves more flexibly.
What are the three conditions for continuity at a point?
A function is continuous at a point if:
The function is defined at the point.
The limit exists at the point.
The function's value equals the limit at that point.
What is the chain rule in differentiation?
The chain rule is used to differentiate composite functions.
What is the difference between a definite and an indefinite integral?
A definite integral has limits of integration and gives a number (accumulated quantity), while an indefinite integral represents a family of antiderivatives with a "+C" constant.
What is the difference between convergence and divergence of a series?
A series converges if the sum approaches a finite value. It diverges if the sum increases without bound or oscillates.
What does a vector-valued function represent?
A vector-valued function assigns a vector to each value of the parameter (usually time), often representing position, velocity, or acceleration in space.
What is the difference between one-sided and two-sided limits?
A one-sided limit only approaches from one direction (left or right), while a two-sided limit considers both sides. A two-sided limit exists only if both one-sided limits are equal.
What is implicit differentiation and when is it used?
Implicit differentiation is used when a function is not explicitly solved for one variable. You differentiate both sides of the equation with respect to x, treating y as a function of x.
What is the Fundamental Theorem of Calculus (conceptually)?
The Fundamental Theorem of Calculus connects differentiation and integration. It states that the integral of a function over an interval can be found using its antiderivative.
What is the purpose of the Ratio Test?
The Ratio Test determines convergence by analyzing the limit of the ratio of successive terms. If the limit is less than 1, the series converges.
What is a polar coordinate system? How is it different from Cartesian?
The polar coordinate system describes points using a radius r and angle θ, rather than x- and y-coordinates. It's useful for circular or spiral shapes.
What is a removable discontinuity vs. a jump discontinuity?
A removable discontinuity occurs when a "hole" exists that could be filled to make the function continuous. A jump discontinuity is when the left- and right-hand limits exist but are not equal.
What does it mean for a function to be differentiable?
A function is differentiable at a point if it has a defined derivative there. This means it must be continuous and have no sharp corners or vertical tangents at that point.
What is a u-substitution and why is it used?
u-substitution is a method used to simplify integration by temporarily replacing a part of the integrand with a single variable, making the integral easier to evaluate.
What is a Taylor Series and what does it approximate?
A Taylor Series is an infinite series that represents a function as a sum of its derivatives evaluated at a point. It gives a polynomial approximation of a function.
What is arc length and how is it conceptually different from distance?
Arc length measures the actual length along a curve, not just the distance between endpoints. It’s like unrolling a curved path and measuring it straight.
What is L’Hôpital’s Rule and when is it applicable?
L’Hôpital’s Rule allows you to evaluate indeterminate limits (like 0/0 or ∞/∞) by differentiating the numerator and denominator and then taking the limit again.
What is logarithmic differentiation and why is it helpful?
Logarithmic differentiation involves taking the natural log of both sides of an equation to simplify differentiation, especially useful for products, quotients, and variables in exponents.
What does it mean for a region to be revolved to create a solid of revolution?
A solid of revolution is formed when a region is rotated around a line (axis), and integration is used to find its volume using methods like disks, washers, or shells.
What does interval of convergence mean for a power series?
The interval of convergence is the set of all x-values for which a power series converges. It includes the radius of convergence and checks endpoint behavior.
How do you interpret velocity, acceleration, and speed in vector functions?
In vector functions:
Velocity is the derivative of position.
Acceleration is the derivative of velocity.
Speed is the magnitude of the velocity vector (a scalar).