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Limits and Derivatives
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The concept (and basic properties) of limits and derivatives
Want a deeper conceptual understanding? Try our interactive lesson!
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
One way to conceptualize a limit is that in the graph above, we can get as close to an output of 2 as we want near x=1.
Focus on the right side of the curve as it approaches the hole, we could find an x so that 1.9<f(x)<2, 1.99<f(x)<2, or even 1.9999<f(x)<2.
The key idea here is that since x→1limf(x)=2, pick a value as close to 2 as you want, and we can find an x-value close enough to 1 so that f(x) is even closer to 2:
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 function that tells you the slope of f(x) at a certain x coordinate.
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 usingMATHthen8:nDeriv(in the following format:dXd(Y1(x))∣X=XTo enter Y1, press
VARSthen scroll toY-VARSand selectFUNCTIONthen Y1.Press
GRAPHto 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)🚫
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