Must f(a) be defined for the limit at a to exist?

No. The limit depends on values near a, not at a.

What is needed for a limit to exist?

f(x) approaches the same number from both the left and the right.

How do you estimate a limit from a table?

Evaluate f(x) on both sides of a, getting closer each time, and see what value they approach.

A straight line joining two points on a curve.

What does the gradient of a curve at a point equal?

The gradient of the tangent at that point.

How is the gradient of a curve a limit?

It is the limit of the chord gradients as the interval h\to0.

Three notations for the first derivative?

f'(x), \frac{dy}{dx} and, in context, \frac{dV}{dr} or \frac{ds}{dt}.

What is the gradient function?

The derivative, which gives the gradient of the curve at each x.

What does a negative derivative tell you?

The quantity is decreasing at that instant.

How can you estimate a small change in y?

\frac{dy}{dx}\times the change in x.

Does \frac{dV}{dt}=12 mean the volume is 12?

No. It means the volume is changing at 12 units per unit time at that instant.

5.1 Limits and the derivativeIB Maths: Applications and Interpretation SL: Flashcards

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What does $\lim_{x\to a}f(x)=L$ mean?

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What does $\lim_{x\to a}f(x)=L$ mean?: $f(x)$ gets closer and closer to $L$ as $x$ gets closer to $a$ .
Must $f(a)$ be defined for the limit at $a$ to exist?: No. The limit depends on values near $a$ , not at $a$ .
What is needed for a limit to exist?: $f(x)$ approaches the same number from both the left and the right.
How do you estimate a limit from a table?: Evaluate $f(x)$ on both sides of $a$ , getting closer each time, and see what value they approach.
What is a chord?: A straight line joining two points on a curve.
Gradient of the chord from $x=a$ to $x=a+h$ ?: $\frac{f(a+h)-f(a)}{h}$ .
What does the gradient of a curve at a point equal?: The gradient of the tangent at that point.
How is the gradient of a curve a limit?: It is the limit of the chord gradients as the interval $h\to0$ .
Three notations for the first derivative?: $f'(x)$ , $\frac{dy}{dx}$ and, in context, $\frac{dV}{dr}$ or $\frac{ds}{dt}$ .
What is the gradient function?: The derivative, which gives the gradient of the curve at each $x$ .
Units of $\frac{dV}{dt}$ if $V$ is in litres and $t$ in minutes?: Litres per minute, L min $^{-1}$ .
What does a negative derivative tell you?: The quantity is decreasing at that instant.
How can you estimate a small change in $y$ ?: $\frac{dy}{dx}\times$ the change in $x$ .
Does $\frac{dV}{dt}=12$ mean the volume is $12$ ?: No. It means the volume is changing at $12$ units per unit time at that instant.

Exam questions on 5.1 Limits and the derivative

The function $f$ is defined by $f(x)=\frac{\sqrt{x+9}-3}{x}$ for $x\neq0$ . A table of values gives $f(-0.1)=0.16713$ , $f(-0.01)=0.16671$ , $f(0.01)=0.16662$ and $f(0.1)=0.16621$ (5 s.f.). Use your GDC where needed.
Use your GDC to evaluate $f(0.0001)$ and $f(-0.0001)$ , and hence write down the limit as $x\to0$ to $3$ significant figures.2 marks
A ball is thrown upwards. Its height above the ground is $h(t)=20t-5t^2$ metres, $t$ seconds after it is thrown. Use your GDC or calculator.
Find the average rate of change of $h$ between $t=1$ and $t=1.001$ and hence estimate $h'(1)$ , stating its units.2 marks
The volume of water in a tank is $V$ litres, $t$ minutes after a tap is opened. The model gives $V(5)=60$ , $V(5.1)=61.18$ and $\frac{dV}{dt}=12$ when $t=5$ .
Interpret $\frac{dV}{dt}=12$ when $t=5$ , including units, and use it to estimate the volume of water when $t=5.5$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).