AP Calculus AB - Ultimate Guide (copy)

Limits

The value a function approaches as the variable within the function gets closer to a specific value.

Ways to Find Limits

Various methods to determine the limit of a function, including graph analysis, estimation from a table, and algebraic manipulation.

Squeeze Theorem

States conditions for a function squeezed between two others to have the same limit as those functions.

Continuity

Describes the behavior of a function at a particular point or over an interval, including jump, essential, and removable discontinuities.

Removing Discontinuities

Process of redefining a function to eliminate discontinuities, often by factoring out common roots.

Asymptotes

Lines that a function approaches but never crosses, including vertical and horizontal asymptotes.

Intermediate Value Theorem (IVT)

Ensures the existence of a value within an interval for a continuous function.

Derivative

Represents the rate of change of a function, calculated through difference quotients or limits.

Derivative Rules

Guidelines for finding derivatives efficiently, including constant, power, product, and quotient rules.

Chain Rule

Method for finding the derivative of composite functions by combining derivatives of inner and outer functions.

Implicit Differentiation

Technique to find derivatives of functions where one variable cannot be isolated, often involving the product rule.

Inverse Function Differentiation

Formula to find the derivative of an inverse function by taking the reciprocal of the derivative at the corresponding y value.

Contextual Applications of Differentiation

Applying derivatives to interpret slopes, velocities, accelerations, and non-motion changes in real-world scenarios.

Related Rates

Problems where the change of one thing is related to another, requiring differentiation and substitution to find the rate of change.

Linearization

Using differentials to approximate the value of a function, where f(x + Δx) ≈ f(x) + f’(x)Δx.

L’Hospital’s Rule

A method to evaluate indeterminate limits of the form 0/0 or ∞/∞ by taking the derivative of the numerator and denominator successively.

Mean Value Theorem (MVT)

Links the average rate of change and the instantaneous rate of change, stating that the slope of the tangent line equals the slope of the secant line at some point.

Extreme Value Theorem

Asserts that a continuous function on a closed interval must have both a maximum and a minimum value.

Intervals of Increase and Decrease

Using the first derivative to determine where a function is increasing (f’(x) > 0) or decreasing (f’(x) < 0) by finding critical points.

Relative Extrema

Points where the first derivative changes sign, indicating relative maxima or minima.

Candidate’s Test & Absolute Extrema

Method to find absolute extrema by considering endpoints and critical numbers of a function.

Function Concavity

Determined by the second derivative, where f”(x) > 0 indicates concave up and f”(x) < 0 indicates concave down.

Integral & Area Under A Curve

The antiderivative representing total change, with the definite integral showing the area under a curve and the x-axis.

Riemann & Trapezoidal Sums

Methods to estimate areas under curves using rectangles or trapezoids, with left, right, midpoint, and trapezoidal sums for accuracy.

Tabular Riemann Sums

Utilizing tabular data to calculate Riemann Sums for functions given in a table format.

Trapezoids

A geometric shape with four sides where the sum of the lengths of the two parallel sides is multiplied by the height and divided by 2 to find the area.

Fundamental Theorem of Calculus

States that the integral of a function can be found by evaluating its antiderivative at the upper and lower bounds of the integral and taking the difference.

Antiderivatives

The reverse process of differentiation, where the power rule is commonly used to find the antiderivative of a function.

Constant of Integration (+C)

Represents the unknown constant that arises when finding the antiderivative of a function.

Definite Integral

An integral with specified upper and lower bounds, used to find the area under a curve between two points.

U-Substitution

A technique in integration where a substitution is made to simplify the integrand, often involving choosing a term as "u" and its derivative to replace other terms.

Slope Fields

Graphical representations showing the slopes of solutions to a differential equation at different points.

Differential Equations

Equations involving derivatives that model the relationship between variables and require solving for the original function.

Average Value of Functions

Calculated by finding the integral of a function over an interval and dividing by the length of the interval.

Area Between Two Curves

The area enclosed by two functions can be found by subtracting the integral of the lower function from the integral of the upper function over a given interval.

Volume by Cross Sectional Area

The volume of a 3D object obtained by rotating a 2D shape around an axis can be found by integrating the cross-sectional area formula over the height range.