10.04 First principles
Interactive practice questions
Consider the function $f\left(x\right)=x^3$f(x)=x3
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=b−a |
$\frac{f\left(b\right)-f\left(a\right)}{b-a}$f(b)−f(a)b−a |
|---|---|---|---|
| $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.
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 | $\editable{}$ |
We can therefore deduce that the instantaneous rate of change of $f\left(x\right)$f(x) at any point $x$x is:
Consider the function $f\left(x\right)=x^2$f(x)=x2
Consider the function $f\left(x\right)=2x^2$f(x)=2x2
Consider the function $f\left(x\right)=3x+2$f(x)=3x+2.