MATH

Hence

The value or equation from the previous part is required to complete the current part.

Hence or otherwise

Information from the prior part can be used for the current part, but other methods may also be applicable.

Tangent Line

A line that touches a curve at a point and has the same slope as the curve at that point, determined using a derivative.

Normal Line

A line perpendicular to the tangent line, passing through the same point of tangency.

Point of Tangency

The point where the tangent line, normal line, and the curve intersect.

Minimum / Maximum

To find, set the derivative of the function to zero.

Parallel

Two lines with the same slope, often indicating derivative usage.

Perpendicular

Two lines with slopes that are opposite reciprocals of each other.

Equation of the line

Usually in point-slope form; requires a point on the line and its slope.

One, two or no real solutions

Look for quadratic forms and use the discriminant to determine the number of solutions.

Increasing / Decreasing

Determine the sign of the derivative at a point or over an interval.

Concave Up / Concave Down

Check the second derivative's sign at a point or over an interval.

First Derivative Test

A method to determine maximums or minimums by analyzing sign changes around critical values.

Second Derivative Test

Evaluating critical points in the second derivative to determine their concavity and potential extremum.

Critical Values

Points where the first derivative is zero or undefined, indicating potential maximum or minimum points.

Points of Inflection

Locations where the second derivative changes sign, indicating a change in concavity.

Probability Distribution

The sum of the probabilities in a distribution must equal one.

Expected Value

Calculated by multiplying values in a probability distribution by their probabilities and summing the results.

Gradient

A synonym for slope, often used to refer to tangent lines.

Inverse Functions

Functions that swap their domain and range; reflections over the line y = x.

y in terms of x

An equation with y isolated on one side, expressed solely in terms of x.

Area of Shaded Region

Calculated by integrating the difference between top and bottom functions over the designated boundaries.

Probability 'Given that'

Refers to conditional probabilities where the denominator is the probability of the specified condition.

Depreciates / Appreciates

Describes a decrease or increase in value, respectively, often in growth or decay models.

Changes direction

In velocity problems, this occurs when velocity is zero and its sign transitions.

Speeds up / slows down

Acceleration and velocity must have opposite signs to slow down, same signs to speed up.

At rest

Refers to a state when velocity equals zero.

Common Tangent

Occurs when the derivatives of two functions are equal at a point of shared tangency.