MATH

• Show that: An answer is provided and it is up to you to justify it. Often, these values are

followed by “hence” statements and need to be used in later parts of the problem.

• Hence: The value or equation determined in the prior part of the problem is needed to complete your current part of the problem.

• Hence or otherwise: You can use the information in the prior part to work the current part, but it is not necessarily the only way to do so.

• Tangent Line: You probably need to either find or use a derivative to determine the slope of the tangent line.

• Normal Line: This line is perpendicular to the tangent line and passes through the same point of tangency.

• Point of Tangency: The point on the graph where the normal line, the tangent line and the curve all intersect.

• Minimum / Maximum (or Least / Greatest or Smallest / Largest): Set the derivative of whatever you want to maximize or minimize equal to zero.

• Parallel: Two lines have the same slope. Can sometimes indicate derivative usage, but mention of a tangent will likely happen if so.

• Perpendicular: Two lines have opposite reciprocal slopes (change the sign and flip the fraction). Can sometimes indicate derivative usage, but mention of a tangent line will likely happen if so.

• Equation of the line: Point-slope form (gradient-slope form) is usually the default. If you can figure out the coordinates of a point on the line and the slope of the line, you can write the equation directly. Only solve for “y = “ if they require it.

• One, two or no real solutions: If any of these scenarios are mentioned, look to see if the equation can be expressed in a quadratic form. If so, make use of the discriminant.

D > 0 indicates two real solutions

D = 0 indicates one real solution

D < 0 indicates no real solutions

• Increasing / Decreasing: Evaluate your derivative at the location / on the interval to see if it’s positive or negative, respectively.

• Concave Up / Concave Down: Evaluate your second derivative at the location / on the interval to see if it’s positive or negative, respectively.

• First Derivative Test: Used to help determine maximums or minimums based on the change of signs on either side of a critical value. Positive to Negative is a maximum; negative to positive is a minimum.

• Second Derivative Test: Evaluate critical points in the second derivative to see whether the potential max / min is in a concave up / concave down region. If concave down, it’s a max. If concave up, it’s a min.

• Critical Values: Locations where the first derivative is undefined or equal to zero. These are potential max / min locations.

• Points of Inflection: Locations where concavity changes. Finding where the second derivative is equal to zero or undefined gives possible locations of points of inflection.

• Probability Distribution: The sum of the probabilities is always equal to one.

• Expected Value: Multiply the values in a probability distribution by their respective probabilities and add all of the results together.

• Gradient: This is a synonym for slope. Often used in reference to tangent lines, but not always.

• Inverse Functions: The most important thing here is that their domain and range swap completely. If a point (a, b) exists on the original graph, its inverse contains (b, a). The two inverses are also reflections of each other over the line 𝑦 = 𝑥.

• “y in terms of x”: You should have an equation with and then only the variable “x” as an𝑦 = input on the right hand side.

• Area of Shaded Region (from a graph of one or more functions): You will be integrating over the region. The boundaries are based on the points of intersection and you should integrate the “top function - bottom function”. This may contain multiple regions, which means you’ll have to adjust which is “top” or “bottom” as needed.

• Probability “Given that”: This indicates a subgroup or particular focus for a probability question. When you calculate the probability of “this given that”, the denominator is always the probability of “that”.

• Depreciates / Appreciates: Increases or decreases, typically a Decay / Growth model.

• Changes direction: In velocity problems, this happens when velocity is equal to zero and its sign changes from negative to positive or positive to negative.

• Speeds up / slows down: Acceleration and velocity must have opposite signs to slow down. They must have the same sign to speed up.

At rest: Velocity = 0

• Common Tangent: The derivatives of both functions are the same value at that location andshare the same point of tangency.