Continuity and Differentiability: Mixed Practice

Before you start#

Twelve ques­tions from across the chap­ter. For con­ti­nu­ity ques­tions, always check three things side by side: the left limit, the right limit and the value. For deriv­a­tives, say to your­self which rule you are using before you start writ­ing. It stops a lot of care­less slips.

The ques­tions#

1. Find $k$ so that $f(x) = \begin{cases} kx^2, & x \le 2 \\ 3, & x > 2 \end{cases}$ is con­tin­u­ous at $2$.
2. Dis­cuss the con­ti­nu­ity of $f(x) = \lvert x \rvert - \lvert x + 1 \rvert$.
3. Find the points of dis­con­ti­nu­ity of $\displaystyle f(x) = \frac{x^2 + 1}{x^2 - 5x + 6}$.
4. Dif­fer­en­ti­ate $\sin(\cos(x^2))$.
5. Dif­fer­en­ti­ate $\displaystyle \sec^{-1}\frac{1}{2x^2 - 1}$ for $\displaystyle 0 < x < \tfrac{1}{\sqrt{2}}$.
6. Find $\displaystyle \tfrac{dy}{dx}$ if $y^x = x^y$.
7. Find $\displaystyle \tfrac{dy}{dx}$ if $x = \cos\theta + \theta\sin\theta$, $y = \sin\theta - \theta\cos\theta$.
8. Dif­fer­en­ti­ate $(\log x)^{\cos x}$.
9. If $y = 3\cos(\log x) + 4\sin(\log x)$, show $x^2 y'' + x y' + y = 0$.
10. If $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$, show $\displaystyle \tfrac{dy}{dx} = \sqrt{\tfrac{1 - y^2}{1 - x^2}}$. (Put $x = \sin A$, $y = \sin B$.)
11. Dif­fer­en­ti­ate $x^{x^2 - 3} + (x - 3)^{x^2}$ for $x > 3$.
12. Show that $f(x) = \lvert x - 1 \rvert + \lvert x - 2 \rvert$ is con­tin­u­ous every­where but not dif­fer­en­tiable at $1$ and $2$.

Answers to check against#