Skills Checklist | IB Math AAHL

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Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

Track your progress:

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Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

38 Skills Available

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Limits and Derivatives

10 skills

Basic concept of a limit

SL 5.1

The limit x→alimf(x) is the value f(x) approaches as x approaches a.

Slope as a Limit

SL 5.1

The IB may test your understanding of the gradient of the curve as the limit of

m=x2−x1y2−y1

as (x2−x1) goes to zero.

Limit from a graph

SL 5.1

x→1limf(x)=2

x→∞limf(x)=23

x→−∞limf(x)=23

Limit from a table

SL 5.1

Given a table of values:

xf(x)0.91.620.991.91210.9991.9972201……

x→1limf(x)=2

Limit definition of derivative

AHL 5.12

The derivative of f(x) is denoted f′(x) and is given by

f′(x)=h→0limhf(x+h)−f(x)📖

Gradient

SL 5.1

For a curve y=f(x), f′(x) is the gradient or slope.

Graphing a derivative with a GDC

SL 5.1

You can graph f′(x) using the following steps:

Press the Y= key.
In one of the available function lines (e.g. Y_1), enter the expression for f(x).
In another available line (e.g. Y_2), input the derivative function usingMATH then 8:nDeriv( in the following format:
dXd(Y1(x))∣X=X
To enter Y1, press VARS then scroll to Y-VARS and select FUNCTION then Y1.
Press GRAPH to display both the original graph f and the derivative f′.
The graph of f′ may take a little bit longer depending on the original function.

After graphing f′, you may use all the other graphing functions on the calculator (intersect, zero, and value).

Rate of Change

SL 5.1

dxdy is the rate of change of y with respect to x. That is, dxdy tells us how much y changes in response to a change in x.

If y=f(x), then dxdy=f′(x).

Derivative of xⁿ where n is an integer

SL 5.3

f(x)=xn, n∈Z⇒f′(x)=nxn−1📖

Derivatives of sums and scalar multiples

SL 5.3

dxd(af(x))=af′(x)🚫

dxd(f(x)+g(x))=f′(x)+g′(x)🚫

dxd(af(x)+bg(x))=af′(x)+bg′(x)🚫

Differentiation rules

12 skills

Derivative of xⁿ where n is an integer

SL 5.3

f(x)=xn, n∈Z⇒f′(x)=nxn−1📖

Derivatives of sums and scalar multiples

SL 5.3

dxd(af(x))=af′(x)🚫

dxd(f(x)+g(x))=f′(x)+g′(x)🚫

dxd(af(x)+bg(x))=af′(x)+bg′(x)🚫

Derivative of xⁿ where n is rational

SL 5.6

f(x)=xn, n∈Q⇒f′(x)=nxn−1📖

Chain rule

SL 5.6

(g(f(x)))′=g′(f(x))⋅f′(x)🚫

y=g(u) where u=f(x)

dxdy=dudg⋅dxdu📖

Derivative of e^x

SL 5.6

f(x)=ex⇒f′(x)=ex📖

Derivative of ln

SL 5.6

f(x)=lnx⇒f′(x)=x1📖

Product and Quotient rule

SL 5.6

The product and quotient rules are given by

(uv)′=u′v+v′u📖

(vu)′=v2u′v−v′u📖

Derivatives of sin and cos

SL 5.6

f(x)=sinx⇒f′(x)=cosx📖

g(x)=cosx⇒g′(x)=−sinx📖

Derivative of tan(x)

AHL 5.15

f(x)=tanx⇒f′(x)=sec2(x)📖

Derivatives of reciprocal trig functions

AHL 5.15

f(x)=secx⇒f′(x)=secxtanx📖

g(x)=cosecx⇒g′(x)=−cosecxcotx📖

h(x)=cotx⇒h′(x)=−cosec2(x)📖

Derivatives of inverse trig functions

AHL 5.15

f(x)=arcsinx⇒f′(x)=√1−x21📖

g(x)=arccosx⇒g′(x)=√1−x2−1📖

h(x)=arctanx⇒h′(x)=1+x21📖

Derivatives of aˣ and logₐ(x)

AHL 5.15

f(x)=ax⇒f′(x)=axlna📖

g(x)=logax⇒f′(x)=xlna1📖

Tangents and normals

2 skills

Tangent to f(x)

SL 5.4

L:mx+c is tangent to f(x) at x=a means

same ysame y′{f(a)=ma+cf′(a)=m🚫

Using point slope form the equation of the tangent is:

y−f(a) ⇒y=m⋅(x−a)🚫 =mx−ma+f(a)🚫

Normal to f(x)

SL 5.4

The normal to f(x) at x=a is the line that passes through (a,f(a)) and is perpendicular to the tangent:

mn⋅mt=−1⇔mn =−mt1🚫 =−f′(a)1🚫

Using point slope form the equation of the tangent is:

y−f(a) ⇒y=mn⋅(x−a)🚫 =mnx−mna+f(a)🚫

Applications of the First Derivative

3 skills

Stationary points & Increasing/Decreasing Regions

SL 5.2

f′(x)⎩⎪⎨⎪⎧<0⇔f decreasing=0⇔f stationary>0⇔f increasing🚫

Maxima & Minima

SL 5.7

Stationary points are often local extrema.

If f′(a)=0, f is decreasing to the left of a (f′(x)<0), and f is increasing to the right of a (f′(x)>0), then (a,f(a)) is a local minimum.

If f′(a)=0, f is increasing to the left of a (f′(x)<0), and f is decreasing to the right of a (f′(x)>0), then (a,f(a)) is a local maximum.

Optimisation

SL 5.8

Optimisation problems require you to find a minimum or maximum value by producing a function f(x), taking its derivative, solving f′(x)=0, and confirming which stationary point(s) are minima or maxima.

Second Derivatives and Applications

5 skills

Second Derivative

SL 5.7

The derivative of the derivative of a function is its second derivative:

f′′(x)=(f′(x))′🚫

dx2d2y=dxd(dxdy)=y′′🚫

Concavity

SL 5.7

We determine concavity by the sign of f′′:

f′′{>0⇔f concave up<0⇔f concave down🚫

Classifying stationary points using the second derivative

SL 5.7

At a stationary point (f′(a)=0),

If f′′(a)>0, then f has a local minimum at x=a.
If f′′(a)<0, then f has a local maximum at x=a.

Using the second derivative to classify a stationary point is often called the second derivative test.

Inflexion Points

SL 5.7

Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫

Graphs of f, f' and f''

SL 5.7

When f′ crosses the x-axis f has a maximum (f′′<0) or minimum (f′′>0)
When f′′ crosses the x-axis, f has an inflexion point.

n^th Derivative

1 skill

n^th Derivative

AHL 5.12

The nth derivative of f is

f(n)(x)=f′′′′…′′n times(x)🚫

dxndny=dxd(dxd(…dxd(y)))n times🚫

6 skills

Related Rates

AHL 5.14

Given three variables x, y, and z,

dzdy=dzdx⋅dxdy.

Hence, given dzdx, we can find an expression for dzdy by calculating dxdy.

Implicit Differentiation

AHL 5.14

Implicit differentiation is when we differentiate both sides of an equation. It is helpful when we have an equation that cannot be simplified to y=f(x).

Related rates with implicit differentiation

AHL 5.14

Since dtdy=dxdy⋅dtdx, you may be asked to use implicit differentiation to find dxdy, then with a given dtdx and point, you can find dtdy.

Volume related rates

AHL 5.14

Given the time rate of change of radius, length, height, or width of a three dimensional object, you may find the time rate of change of volume by taking the derivative of the volume equation.

Distance related rates

AHL 5.14

Let L be the distance from the origin of a point with coordinates (x,y). Then, given dtdx and dtdy, we can find dtdL at a given point (x,y).

Angle related rates

AHL 5.14

Using a given rate of change dtdx and trigonometry, we can calculate dxdθ, which can be used to find dtdθ.

L'Hôpital's rule

1 skill

L'Hôpital's rule

AHL 5.12

If g(a)f(a)=00 or ∞∞, then:

x→alimg(x)f(x)=x→alimg′(x)f′(x)🚫