5.12 Continuity, differentiability and first principles questions for IB Maths AA

Find $f^{\mathrm{\prime }}(x)$ and $f^{\mathrm{\prime }\mathrm{\prime }}(x)$.

Prove by mathematical induction that

Copy and complete the following table.

Show that $Q(h)=h^2+4h+4$.

Hence find the gradient of the graph of $f$ at $x=2$, explaining how your answer relates to the values in the table.

Use this definition to show that the derivative of $x^2$ is $2x$.

Show that $(x+h{)}^3-x^3=h[(x+h{)}^2+(x+h)x+x^2]$.

Hence find the derivative of $x^3$ from first principles.

Let $n\in {\mathbb{Z}}^{+}$, $n\ge 2$, and let $a,b\in \mathbb{R}$. By expanding the right-hand side, show that

Hence prove that the derivative of $x^n$ is $n{x}^{n-1}$, for $n\in {\mathbb{Z}}^{+}$, $n\ge 2$.

Let $f$ and $g$ be polynomials and let $P(x)=f(x)g(x)$.

Show that $P(x+h)-P(x)=f(x+h)[g(x+h)-g(x)]+g(x)[f(x+h)-f(x)]$.

You may assume that ${\displaystyle \underset{h\to 0}{\lim }f(x+h)=f(x)}$ for any polynomial $f$.

Hence show that $P^{\mathrm{\prime }}(x)=f(x)g^{\mathrm{\prime }}(x)+g(x)f^{\mathrm{\prime }}(x)$.

Use the result of part (f) to show that the derivative of ${[f(x)]}^2$ is $2f(x)f^{\mathrm{\prime }}(x)$.

Use mathematical induction to prove that, for all $n\in {\mathbb{Z}}^{+}$, the derivative of ${[f(x)]}^n$ is $n{[f(x)]}^{n-1}{f}^{\mathrm{\prime }}(x)$.

Let $q(x)={(x^2-3x+1)}^5$. Use the result of part (g)(ii) to find $q^{\mathrm{\prime }}(2)$, and check your answer using your GDC.

Show that $f^{\mathrm{\prime }}(x)=\sqrt{2}{\mathrm{e}}^x\sin (x+{\displaystyle \frac{\pi }{4}})$.

Prove by mathematical induction that $f^{(n)}(x)=2^{\frac{n}{2}}{\mathrm{e}}^x\sin (x+{\displaystyle \frac{n\pi }{4}})$, for all $n\in {\mathbb{Z}}^{+}$.

Hence find the first four non-zero terms of the Maclaurin series for $f(x)$.

Hence find ${\displaystyle \underset{x\to 0}{\lim }\frac{{\mathrm{e}}^x\sin x-x-x^2}{x^3}}$.

Find, in terms of $m$, the coefficient of $x^{4m+1}$ in the Maclaurin series for $f(x)$, where $m\in \mathbb{N}$.

Find the value of $a$ and the value of $b$.

Find the greatest angle between the curved section of the ramp and the horizontal.

Consider the case $p(x)=2x^3-5x^2+4x-7$.

Find $p^{\mathrm{\prime }}(x)$, $p^{\mathrm{\prime }\mathrm{\prime }}(x)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(x)$.

Find $p(0)$, $p^{\mathrm{\prime }}(0)$, $p^{\mathrm{\prime }\mathrm{\prime }}(0)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)$. Hence verify that ${\displaystyle \frac{p^{(k)}(0)}{k!}}$ is equal to the coefficient of $x^k$ in $p(x)$, for $k=0,1,2,3$.

Now consider the general polynomial $p$. Show that $p^{(k)}(0)=k!a_k$, for $k=0,1,\dots ,n$.

Hence write $p(x)$ in terms of $x$ and the values $p^{(k)}(0)$, for $k=0,1,\dots ,n$.

Use part (c) with $p(x)=(1+x{)}^n$ to show that the coefficient of $x^k$ in the expansion of $(1+x{)}^n$ is ${\displaystyle \frac{n!}{k!(n-k)!}}$.

Fix a value of $x$, and let $q(h)=p(x+h)$, a polynomial in $h$ of degree $n$.

By applying part (c) to $q$, show that

Hence show that ${\displaystyle \underset{h\to 0}{\lim }{\displaystyle \frac{p(x+h)-p(x)}{h}}=p^{\mathrm{\prime }}(x)}$.

Let $p(x)=x^4-2x^3$. Use the result in part (e) with $x=1$ to express $p(1+h)$ as a polynomial in $h$. Hence find the exact value of $p(1.1)$, and check your answer using your GDC.

The polynomial $p$ is said to have a repeated root at $x=a$ if $p(x)=(x-a{)}^{2}r(x)$ for some polynomial $r$.

Show that $p$ has a repeated root at $x=a$ if and only if $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

Find the values of $k$ for which the equation $x^3-3x+k=0$ has a repeated root. For each value of $k$, write down all the roots of the equation.