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This topic is contained in pages $7-11$ of the textbook.

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The rate of change of a variable with respect to another variable can be represented using differential calculus notation → $\dfrac{dy}{dx}=$ rate of change of $y$ with respect to $x$.

This is useful when considering the related quantities $t$, $s$, $v$, and $a$.

• $v=\dfrac{ds}{dt}$ $=$ rate of change of displacement, $s$, with respect to time, $t$

  → $v=\lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}= \frac {ds}{dt}$

• $a=\dfrac{dv}{dt}$ $=$ rate of change of velocity, $v$, with respect to time, $t$

  → $a=\lim_{u \to v} \frac{v-u}{t}= \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}= \frac {dv}{dt}$

  • $a=\dfrac{d^2s}{dt^2}$ $=$ rate of change of the rate of change of displacement, $s$, with respect to time, $t$

    → $a=\frac{dv}{dt}=\frac{d}{dt} (\frac{ds}{dt})=\frac{d^2s}{dt^2}$

These quantities can also be linked by integration.

• $s(t)=\int v(t) \space dt$
• $v(t)=\int a(t) \space dt$

The gradient of a curve (or a straight line) on a motion–time graph represents instantaneous rate of change, and can be found by differentiation.

The area under a line on a graph can be found by integration.

→ The gradient of a curve (or a straight line) on a displacement–time graph is the instantaneous velocity.

→ The gradient of a curve (or a straight line) on a velocity–time graph is the instantaneous acceleration.

→ The area under an acceleration–time graph between limits is the change in velocity.

→ The area under a velocity–time graph between limits is the displacement.

The linear equations of motion can be derived using these calculus methods.