SL 5.1 Derivative interpreted as gradient function and as rate of change.
Content
Derivative interpreted as gradient function and as rate of change.
Understandings:
- Rates of Change
- RATE OF CHANGE (OR GRADIENT) IN A STRAIGHT LINE
- RATE OF CHANGE (OR GRADIENT) ΙΝ Α CURVE
- Average Rates of Change: Motion
- average velocity
- Instantaneous rate of change
- THE GRADIENT OF A TANGENT
Guidance, clarification and syllabus links
- Forms of notation: \(\frac{dy}{dx},f'(x),\frac{dV}{dr}or \frac{ds}{dt}\) for the first derivative.
- Informal understanding of the gradient of a curve as a limit.
Question
[with GDC]
Let \(f(x)=\frac{x^{3}+1}{\sin x}\)
(a) Find \(f'(x)\).
(b) Find the gradient of the curve \(y=f(x)\)
(ⅰ) at \(x=\frac{\pi }{4}\) (ⅰⅰ) at \(x=1 rad\).
Answer/Explanation
Ans
(a) \(f'(x)=\frac{3x^{2}\sin x-(x^{3}+1)\cos x}{\sin ^{2}x}\)
(b) Directly by GDC (i) \(f'(\frac{\pi }{4})\cong 0.518\) (ii) \(f'(1)\cong 2.04\)
[Notice: the exact value for (i) is \(f'(\frac{\pi }{4})=\frac{3\pi ^{2}}{16}\sqrt{2}-\frac{\pi ^{3}+64}{64}\sqrt{2}]\)