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Rates of Change

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Average Rate of Change (Slope of a Secant Line): For any function y=f(x), the average rate of change over the interval [x1, x2] is calculated by Δy/Δx.

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Secant Line: A line that intersects two or more points on a curve.

Tangent Line: A line that touches a curve at a point.

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Instantaneous Rate of Change (Slope of the Tangent Line at a Fixed Value): The value that the average rate of change approaches as Δx approaches zero.  

Limits

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Limit: A single value that f(x) approaches as a given value of x is approached.

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For certain functions, as an x value is approached, a limit does not exist.

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Limit laws can simplify limit calculations.

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The limits of rational functions and polynomials can be found by replacing x with c. Factoring may be required for certain rational functions that result in a denominator of zero when x is replaced with c.

Limits (Precise)

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Finding Deltas.png
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The precise definition of a limit involves the variables ε (epsilon) and δ (delta).

One-Sided Limits

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Limits can be one-sided.

Special Limit sin x / x

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A special property applies to the limit of sin x / x as x → 0.

Continuity

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Discontinuous Graph.png
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Continuous Function: A function that is continuous at every point in its domain.

Discontinuous Function: A function that is discontinuous at one or more points.

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Continuity Test: The criteria that a function must meet to be continuous.

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Removable Discontinuity: A point on a discontinuous function that can have its definition changed to make a new continuous function.

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Extending the domain of f to include x = c. The extension creates a new function that is continuous at x = c.

Limits and Continuity (Trigonometric Functions)

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Trigonometric functions may be involved in questions related to limits and continuity.

Limits (Infinity)

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Limits can involve infinity.

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Dominant Terms: The terms that result from polynomial division. Dominant terms can be used to predict a function’s behavior.

Rational functions and their asymptotes are frequently associated with limits involving infinity. A rational function’s asymptotes can be determined using precalculus methods.

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Graphs may need to be sketched based on conditions.

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Limits of differences (infinity) can be found by multiplying and dividing by the conjugate.

Derivative at a Point

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Finding the Slope of a Function at a Given Point.png
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The derivative of a function at a point can be used to find both the slope and the instantaneous rate of change of a function.

Derivative as a Function

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Differentiation: Calculating the derivative of a function.

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f’(x) can be graphed with certain restrictions. 

It is helpful to know the graphs of the derivatives of basic functions.

Differentiation Rules

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Derivative rules are necessary for certain scenarios and can be used hasten the speed of differentiation.

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Second-Order Derivative: The derivative of the first derivative.

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Higher-Order Derivatives: The nth derivative of a function.

Derivative (Rate of Change)

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Instantaneous Rates: The limits of average rates.

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Velocity: The derivative of position with respect to time.

Speed: The absolute value of velocity.

Acceleration: The derivative of velocity.

Jerk: The derivative of acceleration.

Derivatives of Trigonometric Functions

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The derivatives of trigonometric functions can be found using the derivative chart.

Chain Rule

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Chain Rule: A method of finding the derivative of a composite function.

Implicit Differentiation

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Implicit Relations: Equations involving the variables x and y. Implicit relations require applying a y’ tracker to the derivative of all y terms. The terms with a y’ tracker are then separated to one side of the equation, and then the terms without y’ trackers are divided by the terms with the y’ trackers.

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Normal Line: The line perpendicular to the tangent line.

Derivatives of Inverse and Logarithmic Functions

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Special rules can be applied to find the derivative of the inverse of a function, as well as functions containing logarithms and exponents.

Derivatives of Inverse Trigonometric Functions

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Special rules can be applied to find the derivative of an inverse trigonometric function.

Linearization and Differentials

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Linearization: An approximation of a more complicated function.

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Differentials: The variables dx and dy.

Extreme Values on Closed Intervals

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Extreme Values: Maximum and minimum values.

Absolute Extreme Values: The largest value and smallest value found on an interval. A function that is continuous over a finite closed interval has both an absolute maximum and an absolute minimum value on the interval.

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Local Extreme Values: The largest and smallest values within a segment of the interval. More than one local maximum and minimum can exist.

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Critical Point: An interior point of the domain of a function f where f’ is zero or undefined.

Mean Value Theorem

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Mean Value Theorem: States that there is a point where the tangent line is parallel to the secant line that joins A and B. 
f(x) must be continuous over a closed interval and differentiable on the interval’s interior.

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Some questions may request plotting the zero of a function and its derivative on a number line.

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Derivatives can be used to determine their corresponding function.

Monotonic Functions and the First Derivative Test

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