The basic idea of a limitĀ āxāalimāf(x)āĀ from tables and graphs, slope as a limit, rate of change and gradient, derivatives ofĀ āxn,Ā
The limitĀ āxāalimāf(x)āĀ is the valueĀ āf(x)āĀ approaches asĀ āxāĀ approachesĀ āa.
The IB may test your understanding of the gradient of the curve as the limit of
asĀ ā(x2āāx1ā)āĀ goes to zero.
One way to conceptualize a limit is that in the graph above, we can get as close to an output ofĀ ā2āĀ as we want nearĀ āx=1.
Focus on the right side of the curve as it approaches the hole, we could find anĀ āxāĀ so thatĀ ā1.9<f(x)<2,Ā Ā ā1.99<f(x)<2,āĀ or evenĀ ā1.9999<f(x)<2.
The key idea here is that sinceĀ āxā1limāf(x)=2,Ā pick a value as close toĀ ā2āĀ as you want, and we can find anĀ āxā-value close enough toĀ ā1āĀ so thatĀ āf(x)āĀ is even closer toĀ ā2:
Given a table of values:
For a curveĀ āy=f(x),Ā Ā āfā²(x)āĀ is the function that tells you the slope ofĀ āf(x)āĀ at a certainĀ āxāĀ coordinate.
You can graphĀ āfā²(x)āĀ using the following steps:
Press the Y= key.
In one of the available function lines (e.g. Y_1), enter the expression forĀ āf(x).
In another available line (e.g. Y_2), input the derivative function usingMATH then 8:nDeriv( in the following format:
To enterĀ āY1ā,Ā press VARS then scroll to Y-VARS and select FUNCTION thenĀ āY1ā.
Press GRAPH to display both the original graphĀ āfāĀ and the derivativeĀ āfā².
The graph ofĀ āfā²āĀ may take a little bit longer depending on the original function.
After graphingĀ āfā²,Ā you may use all the other graphing functions on the calculator (intersect, zero, and value).
ādxdyāāĀ is the rate of change ofĀ āyāĀ with respect toĀ āxā. That is,Ā ādxdyāāĀ tells us how muchĀ āyāĀ changes in response to a change inĀ āx.
IfĀ āy=f(x),Ā thenĀ ādxdyā=fā²(x).
Nice work completing Limits and Derivatives, here's a quick recap of what we covered:
Exercises checked off
The basic idea of a limitĀ āxāalimāf(x)āĀ from tables and graphs, slope as a limit, rate of change and gradient, derivatives ofĀ āxn,Ā
The limitĀ āxāalimāf(x)āĀ is the valueĀ āf(x)āĀ approaches asĀ āxāĀ approachesĀ āa.
The IB may test your understanding of the gradient of the curve as the limit of
asĀ ā(x2āāx1ā)āĀ goes to zero.
One way to conceptualize a limit is that in the graph above, we can get as close to an output ofĀ ā2āĀ as we want nearĀ āx=1.
Focus on the right side of the curve as it approaches the hole, we could find anĀ āxāĀ so thatĀ ā1.9<f(x)<2,Ā Ā ā1.99<f(x)<2,āĀ or evenĀ ā1.9999<f(x)<2.
The key idea here is that sinceĀ āxā1limāf(x)=2,Ā pick a value as close toĀ ā2āĀ as you want, and we can find anĀ āxā-value close enough toĀ ā1āĀ so thatĀ āf(x)āĀ is even closer toĀ ā2:
Given a table of values:
For a curveĀ āy=f(x),Ā Ā āfā²(x)āĀ is the function that tells you the slope ofĀ āf(x)āĀ at a certainĀ āxāĀ coordinate.
You can graphĀ āfā²(x)āĀ using the following steps:
Press the Y= key.
In one of the available function lines (e.g. Y_1), enter the expression forĀ āf(x).
In another available line (e.g. Y_2), input the derivative function usingMATH then 8:nDeriv( in the following format:
To enterĀ āY1ā,Ā press VARS then scroll to Y-VARS and select FUNCTION thenĀ āY1ā.
Press GRAPH to display both the original graphĀ āfāĀ and the derivativeĀ āfā².
The graph ofĀ āfā²āĀ may take a little bit longer depending on the original function.
After graphingĀ āfā²,Ā you may use all the other graphing functions on the calculator (intersect, zero, and value).
ādxdyāāĀ is the rate of change ofĀ āyāĀ with respect toĀ āxā. That is,Ā ādxdyāāĀ tells us how muchĀ āyāĀ changes in response to a change inĀ āx.
IfĀ āy=f(x),Ā thenĀ ādxdyā=fā²(x).
Nice work completing Limits and Derivatives, here's a quick recap of what we covered:
Exercises checked off