The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, using the derivative definition f′(x)=h→0limhf(x+h)−f(x),
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.
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:
Given a table of values:
The derivative of f(x) is denoted f′(x) and is given by
Expand (x+h)3=x3+3x2h+3xh2+h3, so
Canceling terms:
Now divide through by h, since h=0 in a limit:
For a curve y=f(x), f′(x) is the function that tells you the slope of f(x) at a certain x coordinate.
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).
Nice work completing Limits and Derivatives, here's a quick recap of what we covered:
Exercises checked off
The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, using the derivative definition f′(x)=h→0limhf(x+h)−f(x),
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.
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:
Given a table of values:
The derivative of f(x) is denoted f′(x) and is given by
Expand (x+h)3=x3+3x2h+3xh2+h3, so
Canceling terms:
Now divide through by h, since h=0 in a limit:
For a curve y=f(x), f′(x) is the function that tells you the slope of f(x) at a certain x coordinate.
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).
Nice work completing Limits and Derivatives, here's a quick recap of what we covered:
Exercises checked off