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9.03 Differentiation from first principles

Interactive practice questions

Consider the function $f\left(x\right)=x^3$f(x)=x3

a

By filling in the table of values, complete the limiting chord process for $f\left(x\right)=x^3$f(x)=x3 at the point $x=3$x=3.

$a$a $b$b $h=b-a$h=ba

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$f(b)f(a)ba

$3$3 $3.5$3.5 $0.5$0.5 $\editable{}$
$3$3 $3.1$3.1 $\editable{}$ $\editable{}$
$3$3 $3.05$3.05 $\editable{}$ $\editable{}$
$3$3 $3.01$3.01 $\editable{}$ $\editable{}$
$3$3 $3.001$3.001 $\editable{}$ $\editable{}$
$3$3 $3.0001$3.0001 $\editable{}$ $\editable{}$

Provide answers up to four decimal places when required.

b

The limiting chord process has been used to calculate the instantaneous rate of change for each value of $x$x given in the table. Complete the missing values.

$x$x $1$1 $2$2 $3$3 $4$4 $5$5

Instantaneous rate of change of $f\left(x\right)$f(x) at $x$x.

$3$3 $12$12 $\editable{}$ $48$48 $\editable{}$
c

We can therefore deduce that the instantaneous rate of change of $f\left(x\right)$f(x) at any point $x$x is:

Easy
8min

Consider the function $f\left(x\right)=x^2$f(x)=x2

Easy
6min

Consider the function $f\left(x\right)=2x^2$f(x)=2x2

Easy
9min

Consider the function $f\left(x\right)=3x+2$f(x)=3x+2.

Easy
2min
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Outcomes

ACMMM078

use the Leibniz notation δx and δy for changes or increments in the variables x and y

ACMMM079

use the notation δx/δy for the difference quotient [f(x+h)−f(x)]/h where y=f(x)

ACMMM080

interpret the ratios [f(x+h)−f(x)]/h and δy/δx as the slope or gradient of a chord or secant of the graph of y=f(x)

ACMMM081

examine the behaviour of the difference quotient [f(x+h)−f(x)] / h as h→0 as an informal introduction to the concept of a limit

ACMMM082

define the derivative f′(x) as lim_h→0 [f(x+h)−f(x)]/h

ACMMM083

use the Leibniz notation for the derivative: dy/dx=lim_δx→0 δy/δx and the correspondence dy/dx=f′(x) where y=f(x)

ACMMM084

interpret the derivative as the instantaneous rate of change

ACMMM085

interpret the derivative as the slope or gradient of a tangent line of the graph of y=f(x)

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