DP Mathematics HL Questionbank

• Syllabus

9.6

Sub sections and their related questions

Rolle’s theorem.

• 14M.3ca.hl.TZ0.4b: (i)     Use Rolle’s theorem, applied to $f$, to prove that...
• 14N.3ca.hl.TZ0.4c: $f$ is a continuous function defined on $[a,{\text{ }}b]$ and differentiable on...
• 14N.3ca.hl.TZ0.4d: (i)     Given $g(x) = x - \arctan x$, prove that $g'(x) > 0$, for $x > 0$. (ii)    ...
• 14N.3ca.hl.TZ0.4e: Use the result from part (c) to prove that $\arctan x > x - \frac{{{x^3}}}{3}$, for...
• 14N.3ca.hl.TZ0.4f: Hence show that $\frac{{16}}{{3\sqrt 3 }} < \pi  < \frac{6}{{\sqrt 3 }}$.

Mean value theorem.

• 15M.3ca.hl.TZ0.5a: The mean value theorem states that if $f$ is a continuous function on $[a,{\text{ }}b]$ and...
• 15M.3ca.hl.TZ0.5b: (i)     The function $f$ is continuous on $[a,{\text{ }}b]$, differentiable on...
• 16M.3ca.hl.TZ0.1a: By finding a suitable number of derivatives of $f$, determine the Maclaurin series for $f(x)$...
• 16M.3ca.hl.TZ0.1b: Hence, or otherwise, determine the exact value of...
• 16M.3ca.hl.TZ0.1c: (i)     Use the Lagrange form of the error term to find an upper bound for the absolute value of...
• 16N.3ca.hl.TZ0.2a: By successive differentiation find the first four non-zero terms in the Maclaurin series for...
• 16N.3ca.hl.TZ0.2b: Deduce that, for $n \geqslant 2$, the coefficient of ${x^n}$ in this series is...
• 16N.3ca.hl.TZ0.2c: By applying the ratio test, find the radius of convergence for this Maclaurin series.
• 16N.3ca.hl.TZ0.4a: State the mean value theorem for a function that is continuous on the closed interval...
• 16N.3ca.hl.TZ0.4b: (i)     Find $g(0)$. (ii)     Find $g(h)$. (iii)     Apply the mean value theorem to the...
• 16N.3ca.hl.TZ0.4c: Hence show that, for $h > 0$ $1 - \cos (h) \leqslant \frac{{{h^2}}}{2}$.

Taylor polynomials; the Lagrange form of the error term.

• 08M.3ca.hl.TZ1.5: (a)     Write down the value of the constant term in the Maclaurin series for $f(x)$ . (b)    ...
• 13M.3ca.hl.TZ0.1a: Find the values of ${a_0},{\text{ }}{a_1},{\text{ }}{a_2}$ and ${a_3}$.
• 15N.3ca.hl.TZ0.2b: By further differentiation of the result in part (a) , find the Maclaurin expansion of $f(x)$,...
• 16M.3ca.hl.TZ0.3a: Given that $f(x) = \ln x$, use the mean value theorem to show that, for $0 < a < b$,...
• 16M.3ca.hl.TZ0.3b: Hence show that $\ln (1.2)$ lies between $\frac{1}{m}$ and $\frac{1}{n}$, where $m$,...
• 16N.3ca.hl.TZ0.2a: By successive differentiation find the first four non-zero terms in the Maclaurin series for...
• 16N.3ca.hl.TZ0.2b: Deduce that, for $n \geqslant 2$, the coefficient of ${x^n}$ in this series is...
• 16N.3ca.hl.TZ0.2c: By applying the ratio test, find the radius of convergence for this Maclaurin series.
• 16N.3ca.hl.TZ0.4a: State the mean value theorem for a function that is continuous on the closed interval...
• 16N.3ca.hl.TZ0.4b: (i)     Find $g(0)$. (ii)     Find $g(h)$. (iii)     Apply the mean value theorem to the...
• 16N.3ca.hl.TZ0.4c: Hence show that, for $h > 0$ $1 - \cos (h) \leqslant \frac{{{h^2}}}{2}$.
• 18M.3ca.hl.TZ0.4a: Show that $f'\left( 0 \right) = 0$.
• 18M.3ca.hl.TZ0.4b: By differentiating the above equation twice, show...
• 18M.3ca.hl.TZ0.4c: Hence show that the Maclaurin series for $f\left( x \right)$ up to and including the term...
• 18M.3ca.hl.TZ0.4d: Use this series approximation for $f\left( x \right)$ with $x = \frac{1}{2}$ to find an...

Maclaurin series for ${{\text{e}}^x}$ , $\\sin x$ , $\cos x$ , $\ln (1 + x)$ , ${(1 + x)^p}$ , $P \in \mathbb{Q}$ .

• 12N.3ca.hl.TZ0.4c: Using the Maclaurin series for $\ln (1 + x)$ , show that the Maclaurin series for...
• 08M.3ca.hl.TZ1.5: (a)     Write down the value of the constant term in the Maclaurin series for $f(x)$ . (b)    ...
• 08M.3ca.hl.TZ2.4: (a)     Given that $y = \ln \cos x$ , show that the first two non-zero terms of the Maclaurin...
• 11M.3ca.hl.TZ0.1a: Find the first three terms of the Maclaurin series for $\ln (1 + {{\text{e}}^x})$ .
• 09M.3ca.hl.TZ0.2: The variables x and y are related by $\frac{{{\text{d}}y}}{{{\text{d}}x}} - y\tan x = \cos x$...
• 09N.3ca.hl.TZ0.2: The function f is defined by $f(x) = {{\text{e}}^{({{\text{e}}^x} - 1)}}$ . (a)     Assuming...
• SPNone.3ca.hl.TZ0.1b: (i)     Find the Maclaurin series for $f(x)$ up to and including the term in ${x^4}$ . (ii)...
• 10M.3ca.hl.TZ0.4: (a)     Using the Maclaurin series for ${(1 + x)^n}$, write down and simplify the Maclaurin...
• 10N.3ca.hl.TZ0.3: (a)     Using the Maclaurin series for the function ${{\text{e}}^x}$, write down the first four...
• 14M.3ca.hl.TZ0.1b: Find the first three non-zero terms in the Maclaurin expansion of $f(x)$.
• 13N.3ca.hl.TZ0.4c: Hence determine the minimum number of terms of the expansion of $g(x)$ required to approximate...
• 14M.3ca.hl.TZ0.1a: Show that $f'(x) = g(x)$ and $g'(x) = f(x)$.
• 13N.3ca.hl.TZ0.4b: Use the Maclaurin series of $\sin x$ to show that...
• 16M.3ca.hl.TZ0.3a: Given that $f(x) = \ln x$, use the mean value theorem to show that, for $0 < a < b$,...
• 16M.3ca.hl.TZ0.3b: Hence show that $\ln (1.2)$ lies between $\frac{1}{m}$ and $\frac{1}{n}$, where $m$,...
• 18M.3ca.hl.TZ0.4a: Show that $f'\left( 0 \right) = 0$.
• 18M.3ca.hl.TZ0.4b: By differentiating the above equation twice, show...
• 18M.3ca.hl.TZ0.4c: Hence show that the Maclaurin series for $f\left( x \right)$ up to and including the term...
• 18M.3ca.hl.TZ0.4d: Use this series approximation for $f\left( x \right)$ with $x = \frac{1}{2}$ to find an...

Use of substitution, products, integration and differentiation to obtain other series.

• SPNone.3ca.hl.TZ0.1b: (i)     Find the Maclaurin series for $f(x)$ up to and including the term in ${x^4}$ . (ii)...
• 10N.3ca.hl.TZ0.3: (a)     Using the Maclaurin series for the function ${{\text{e}}^x}$, write down the first four...
• 13N.3ca.hl.TZ0.4c: Hence determine the minimum number of terms of the expansion of $g(x)$ required to approximate...
• 13N.3ca.hl.TZ0.4b: Use the Maclaurin series of $\sin x$ to show that...
• 14N.3ca.hl.TZ0.4b: Hence show that an expansion of $\arctan x$ is...
• 15M.3ca.hl.TZ0.1: The function $f$ is defined by $f(x) = {{\text{e}}^{ - x}}\cos x + x - 1$. By finding a...
• 15N.3ca.hl.TZ0.4d: Hence show that...

Taylor series developed from differential equations.

• 12M.3ca.hl.TZ0.2b: (i)     Show that...
• 08N.3ca.hl.TZ0.1: (a)     Show that the solution of the homogeneous differential...
• SPNone.3ca.hl.TZ0.3a: By finding the values of successive derivatives when x = 0 , find the Maclaurin series for y as...
• 11N.3ca.hl.TZ0.5b: Hence, by repeated differentiation of the above differential equation, find the Maclaurin series...