Chapter 5

Continuity and Differentiability

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (35)

Examples

Exercise 5.1

Exercise 5.2

Exercise 5.3

Exercise 5.4

Exercise 5.5

Exercise 5.6

Exercise 5.6 · Q1 $\frac{dy}{dx}$ when $x=2at^2$ ⁠, $y=at^4$ worked
Exercise 5.6 · Q2 $\frac{dy}{dx}$ when $x=a\cos\theta$ ⁠, $y=b\cos\theta$ worked
Exercise 5.6 · Q3 $\frac{dy}{dx}$ when $x=\sin t$ ⁠, $y=\cos 2t$ worked
Exercise 5.6 · Q4 $\frac{dy}{dx}$ when $x=4t$ ⁠, $y=\frac4t$ worked
Exercise 5.6 · Q5 $\frac{dy}{dx}$ when $x=\cos\theta-\cos2\theta$ ⁠, $y=\sin\theta-\sin2\theta$ worked
Exercise 5.6 · Q6 $\frac{dy}{dx}$ when $x=a(\theta-\sin\theta)$ ⁠, $y=a(1+\cos\theta)$ worked
Exercise 5.6 · Q7 $\frac{dy}{dx}$ when $x=\frac{\sin^3t}{\sqrt{\cos2t}}$ ⁠, $y=\frac{\cos^3t}{\sqrt{\cos2t}}$ worked
Exercise 5.6 · Q8 $\frac{dy}{dx}$ when $x=a\big(\cos t+\log\tan\frac t2\big)$ ⁠, $y=a\sin t$ worked
Exercise 5.6 · Q9 $\frac{dy}{dx}$ when $x=a\sec\theta$ ⁠, $y=b\tan\theta$ worked
Exercise 5.6 · Q10 $\frac{dy}{dx}$ when $x=a(\cos\theta+\theta\sin\theta)$ ⁠, $y=a(\sin\theta-\theta\cos\theta)$ worked
Exercise 5.6 · Q11Show $\frac{dy}{dx}=-\frac yx$ for $x=\sqrt{a^{\sin^{-1}t}}$ ⁠, $y=\sqrt{a^{\cos^{-1}t}}$ worked

Exercise 5.7

Miscellaneous Exercise

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