Limits and Derivatives

0 of 0 exercises completed

The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of xn, and linearity for sums and scalar multiples.

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 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)🚫

Nice work completing Limits and Derivatives, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Differentiation rules

Differentiation

Limits and Derivatives

0 of 0 exercises completed

The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of xn, and linearity for sums and scalar multiples.

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 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)🚫

Nice work completing Limits and Derivatives, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Differentiation rules

Differentiation

Limits and Derivatives

0 of 0 exercises completed

The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of xn, and linearity for sums and scalar multiples.

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 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)🚫

Nice work completing Limits and Derivatives, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free

Differentiation rules

Differentiation

Limits and Derivatives

0 of 0 exercises completed

The basic idea of a limit x→alimf(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of xn, and linearity for sums and scalar multiples.

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 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)🚫

Nice work completing Limits and Derivatives, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free