Skills Checklist | IB Math AASL

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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:

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

Skill Checklist

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

25 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

9 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

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

8 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📖

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.