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1.7 Differentiation

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Question 33
36%

Jordan is attempting to use differentiation from first principles to prove that the rate of change of the displacement of a pendulum, given by s(t)=sin⁡ts(t) = \sin ts(t)=sint, is −1-1−1 at the instant where t=πt = \pit=π.

Jordan's teacher points out that mistakes were made starting in Step 4 of the derivation. The working is shown below.

Step 1: Gradient of chord PQ=sin⁡(π+h)−sin⁡(π)hPQ = \frac{\sin(\pi + h) - \sin(\pi)}{h}PQ=hsin(π+h)−sin(π)​

Step 2: =sin⁡(π)cos⁡(h)+cos⁡(π)sin⁡(h)−sin⁡(π)h= \frac{\sin(\pi)\cos(h) + \cos(\pi)\sin(h) - \sin(\pi)}{h}=hsin(π)cos(h)+cos(π)sin(h)−sin(π)​

Step 3: =sin⁡(π)(cos⁡(h)−1h)+cos⁡(π)(sin⁡(h)h)= \sin(\pi)\left(\frac{\cos(h) - 1}{h}\right) + \cos(\pi)\left(\frac{\sin(h)}{h}\right)=sin(π)(hcos(h)−1​)+cos(π)(hsin(h)​)

Step 4: For the rate of change at t=πt = \pit=π, let h=0h = 0h=0 then

cos⁡(h)−1h=1 and sin⁡(h)h=0 \frac{\cos(h) - 1}{h} = 1 \text{ and } \frac{\sin(h)}{h} = 0 hcos(h)−1​=1 and hsin(h)​=0

Step 5: Hence the rate of change is given by

sin⁡(π)×1+cos⁡(π)×0=0 \sin(\pi) \times 1 + \cos(\pi) \times 0 = 0 sin(π)×1+cos(π)×0=0

Complete Steps 4 and 5 of Jordan's working to correct the proof.

[3]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.

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