IB Math General Study Guide

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, (y - y1 = m(x - x1)) 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 and share the same point of tangency.

Provided: All three side lengths of a triangle.

Target: Angle measures.

Formula: $m∠C = \cos^{-1}\left( \frac{a^2 + b^2 - c^2}{2ab} \right)$

Note: In the case of only knowing one angle for the Sine Rule / Law of Sines, the ambiguous case may need to be considered. Understand that there may be two possible “correct” triangles.

I’m not going to paste this on every definition but just know

Provided: Two side lengths and their included angle in a triangle.

Target: Third Side Length.

Formula: $c^2 = a^2 + b^2 - 2ab \cos(C)$

Provided: A corresponding pair of side and angle as well as a second angle measure.

Target: The corresponding side to the given angle.

Formula: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$

Provided: A corresponding pair of side and angle as well as a second side length.

Target: The corresponding angle measure.

Formula: $\frac{\sin\left(A\right)}{a}=\frac{\sin\left(B\right)}{b}$

Provided: A point and a slope

Target: Equation of a line

Formula: $y-y_1=m\left(x-x_1\right)$

Provided: A quadratic equation

Target: Solutions / x-intercepts / zeros / roots

Formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Q: What mode does the calculator default to? Degrees or Radians?

A: Radians

Q: What would have to be included in a problem for radian / degree mode to matter?

A: Using sine, cosine or tangent in the operations directly.

Q: Which of the following probability commands do we not use in this course? Binomcdf,

binompdf, normcdf, normpdf, or invNorm?

A: We do not use normpdf. Normpdf would draw the physical curve itself for a normal

distribution, but we do not need to do that.

Q: What information must be available to use binompdf (in order of input)?

A: A set number of trials, a probability of success, and a targeted value.

Q: What information must be available to use binomcdf (in order of input)?

A: A set number of trials, a probability of success and the upper limit of values desired.

Q: In what scenario would you want to make use of binomcdf directly and in what

scenario would you use the “1 - binomcdf” structure? What do you have to be careful

about with the “1 - binomcdf” version?

A: We use binomcdf for “at most” statements and we use “1 - binomcdf” for at least

statements. In the latter case, if you wanted at least X, we use X - 1 in the binomcdf

command.

Q: SL does not provide the formula for $\tan\left(2\theta\right)$. If you were expected to calculate this value, what would you have to do instead?

A: Since $\tan\left(\theta\right)=\frac{\sin\theta}{\cos\theta}$, we can evaluate and and then divide them.

Q: How do you know when to use formulas and commands for a normal distribution,

such as normdcf and invNorm?

A: The problem will explicitly state that you are working with a Normal Distribution.

Q: What are the four components of the normcdf command?

A: Lower Boundary, Upper Boundary, mean, standard deviation

Q: What are the two uses for invNorm and how do they differ?

A: The first use is to identify a particular data value for which you know the probability. In

this case, you list the P(X<#), the mean and the standard deviation. If either the mean or

standard deviation are missing, then you can use the same probability with a mean of 0

and standard deviation of 1 to estimate its z-score instead.

Q: What is different about taking the derivative of $\ln\left(x\right)$ versus integrating its derivative of $\frac{1}{x}$?

A: When integrating, we have to include an absolute value such that $y=\ln\left|x\right|$.

Q: When using a calculator to determine distance travelled between two times, what differs from using it to find displacement?

A: The displacement integrates $v\left(x\right)$ while distance integrates $\left|v\left(x\right)\right|$.