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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
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📖 = included in formula booklet • 🚫 = not in formula booklet
Differentiation
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The limit x→alimf(x) is the value f(x) approaches as x approaches a.
The IB may test your understanding of the gradient of the curve as the limit of
as (x2−x1) goes to zero.
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Given a table of values:
For a curve y=f(x), f′(x) is the gradient or slope.
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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:
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).
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).
The product and quotient rules are given by
L:mx+c is tangent to f(x) at x=a means
Using point slope form the equation of the tangent is:
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The normal to f(x) at x=a is the line that passes through (a,f(a)) and is perpendicular to the tangent:
Using point slope form the equation of the tangent is:
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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.
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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.
The derivative of the derivative of a function is its second derivative:
We determine concavity by the sign of f′′:
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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.
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Inflexion points occur when f′′(x)=0 and f′′(x) changes sign. 🚫
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Given three variables x, y, and z,
Hence, given dzdx, we can find an expression for dzdy by calculating dxdy.
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.