DP Further Mathematics HL Questionbank

Description

The aims of this topic are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations.

Directly related questions

• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.1.hl.TZ0.5: Use the integral test to determine whether or...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11a: Show that...
• 16M.2.hl.TZ0.2a: Use l’Hôpital’s rule to show that...
• 16M.2.hl.TZ0.7b: (i)     Use these results to show that the Maclaurin series for the function \({f_5}(x)\) up to...
• 16M.2.hl.TZ0.7a: Show that (i)     \(\frac{{{\text{d}}{f_n}(x)}}{{{\text{d}}x}} = n{g_n}(x)\); (ii)    ...
• 16M.2.hl.TZ0.4d: (i)     Find the general solution of the differential equation...
• 16M.2.hl.TZ0.4c: (i)     Write down the particular solution passing through the points \((1,{\text{ }} \pm...
• 16M.2.hl.TZ0.4b: (i)     Hence find the particular solution passing through the points...
• 16M.2.hl.TZ0.4a: Find the general solution of the differential equation, expressing your answer in the form...
• 16M.1.hl.TZ0.7b: differentiable at \(x = 0\).
• 16M.1.hl.TZ0.7a: continuous at \(x = 0\);
• 17M.2.hl.TZ0.5b.iii: Hence, by using your calculator to draw two appropriate graphs or otherwise, find the...
• 17M.2.hl.TZ0.5b.ii: Starting with the differential equation, show...
• 17M.2.hl.TZ0.5b.i: Find an integrating factor and hence solve the differential equation, giving your answer in the...
• 17M.2.hl.TZ0.5a.ii: Hence, using integration by parts, show...
• 17M.2.hl.TZ0.5a.i: By considering integration as the reverse of differentiation, show that...
• 17M.1.hl.TZ0.9b.ii: Determine the value of \({I_3}\), giving your answer as a multiple of \({{\text{e}}^{ - 1}}\).
• 17M.1.hl.TZ0.9b.i: Show that, for \(n \in {\mathbb{Z}^ +...
• 17M.1.hl.TZ0.9a: Using l’Hôpital’s rule, show...
• 17M.1.hl.TZ0.7b.iii: Determine the numerical value of \(p\) when \(\mu  = 3\).
• 17M.1.hl.TZ0.7b.ii: Show that \(p = {{\text{e}}^{ - \mu }}f(\mu )\).
• 17M.1.hl.TZ0.7b.i: Write down a series in terms of \(\mu \) for the probability...
• 17M.1.hl.TZ0.7a.ii: By considering derivatives of \(f\), determine the first three non-zero terms of the Maclaurin...
• 17M.1.hl.TZ0.7a.i: Show that \({f^{(4)}}x = f(x)\);
• 17M.1.hl.TZ0.5b.ii: By giving a suitable example, show that it is false.
• 17M.1.hl.TZ0.5b.i: State the converse proposition.
• 17M.1.hl.TZ0.5a: Given that the series \(\sum\limits_{n = 1}^\infty  {{u_n}} \) is convergent, where...
• 15M.2.hl.TZ0.8d: Explain briefly why the same maximum value of error term occurs for \(g(x) = \cos x\) centred...
• 15M.2.hl.TZ0.8c: Hence find the largest number of decimal places to which \(\sin x\) can be estimated for angles...
• 15M.2.hl.TZ0.8b: When using this approximation to find angles between \(130^\circ\) and \(140^\circ\), find the...
• 15M.2.hl.TZ0.8a: Using a Taylor series, find a quadratic approximation for \(f(x) = \sin x\) centred about...
• 15M.2.hl.TZ0.1d: (i)     Find the first three non-zero terms of the Maclaurin series for \(y\). (ii)     Hence...
• 15M.2.hl.TZ0.1c: Solve the differential equation to find an exact value for \(y\) when \(x = 1\).
• 15M.2.hl.TZ0.1b: Explain how Euler’s method could be improved to provide a better approximation.
• 15M.2.hl.TZ0.1a: Using Euler’s method with increments of \(0.2\), find an approximate value for \(y\) when \(x = 1\).
• 15M.1.hl.TZ0.8b: Hence solve the differential equation...
• 15M.1.hl.TZ0.6: Find the interval of convergence of the series...
• 11M.1.hl.TZ0.2a: (i)      Find the range of values of \(n\) for which \(\int_1^\infty  {{x^n}{\rm{d}}x} \)...
• 11M.1.hl.TZ0.2b: Find the solution to the differential...
• 11M.2.hl.TZ0.5a: Find the value of \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{x} - \cot x} \right)\) .
• 11M.2.hl.TZ0.5b: Find the interval of convergence of the infinite...
• 11M.2.hl.TZ0.5c: (i)     Find the Maclaurin series for \(\ln (1 + \sin x)\) up to and including the term in...
• 10M.1.hl.TZ0.5: Given that \(\frac{{{\rm{d}}x}}{{{\rm{d}}y}} + 2y\tan x = \sin x\) , and \(y = 0\) when...
• 10M.2.hl.TZ0.6a: The diagram shows a sketch of the graph of \(y = {x^{ - 4}}\) for \(x > 0\) . By...
• 10M.2.hl.TZ0.6b: Let \(S = \sum\limits_{r = 1}^\infty  {\frac{1}{{{r^4}}}} \) . Use the result in (a) to show...
• 10M.2.hl.TZ0.6c: (i)     Show that, by taking \(n = 8\) , the value of \(S\) can be deduced correct to three...
• 10M.2.hl.TZ0.6d: Now let \(T = \sum\limits_{r = 1}^\infty  {\frac{{{{( - 1)}^{r + 1}}}}{{{r^4}}}} \) . Find the...
• 09M.2.hl.TZ0.1A.b: (i)      Use your answer to (a) to find an approximate expression for the cumulative distributive...
• 09M.1.hl.TZ0.6b: (i)     Sum the series \(\sum\limits_{r = 0}^\infty  {{x^r}} \) . (ii)     Hence, using sigma...
• 09M.2.hl.TZ0.1A.a: Assuming the series for \({{\rm{e}}^x}\) , find the first five terms of the Maclaurin series...
• 09M.2.hl.TZ0.4A.a: Write down the general term.
• 09M.2.hl.TZ0.4A.b: Find the interval of convergence.
• 09M.2.hl.TZ0.4B: Solve the differential equation \((u + 3{v^3})\frac{{{\rm{d}}v}}{{{\rm{d}}u}} = 2v\) , giving...
• 13M.1.hl.TZ0.4b: (i)     Show that the solution \(y = f(x)\) that satisfies the condition...
• 13M.2.hl.TZ0.2b: (i)     Show that the improper integral \(\int_0^\infty  {\frac{1}{{{x^2} + 1}}} {\rm{d}}x\) is...
• 13M.2.hl.TZ0.2d: For the series \(\sum\limits_{n = 0}^\infty  {\frac{{{x^n}}}{{{n^2} + 1}}} \)   (i)    ...
• 13M.1.hl.TZ0.4a: Find the general solution of the differential equation...
• 13M.2.hl.TZ0.2c: (i)     Show that the series...
• 08M.1.hl.TZ0.5: Solve the following differential...
• 08M.2.hl.TZ0.2A.d: By combining your two series, show that...
• 08M.2.hl.TZ0.2A.e: Hence, or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \sec x}}{{x\sqrt x }}\) .
• 08M.2.hl.TZ0.6a: Show that, for \(n \ge 2\) , \({S_{2n}} > {S_n} + \frac{1}{2}\) .
• 08M.2.hl.TZ0.6b: Deduce that \({S_{2m + 1}} > {S_2} + \frac{m}{2}\) .
• 07M.1.hl.TZ0.2b: Calculate the following...
• 08M.2.hl.TZ0.2A.a: Show that \(f''(x) = \frac{{ - 1}}{{1 + \sin x}}\) .
• 08M.2.hl.TZ0.2A.b: Determine the Maclaurin series for \(f(x)\) as far as the term in \({x^4}\) .
• 08M.2.hl.TZ0.2A.c: Deduce the Maclaurin series for \(\ln (1 - \sin x)\) as far as the term in \({x^4}\) .
• 08M.2.hl.TZ0.6c: Hence show that the sequence \(\left\{ {{S_n}} \right\}\) is divergent.
• 07M.1.hl.TZ0.2a: Calculate the following limit \(\mathop {\lim }\limits_{x \to 0} \frac{{{2^x} - 1}}{x}\) .
• 07M.1.hl.TZ0.5: Solve the differential equation \(x\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + 2y = \sqrt {1 + {x^2}}...
• 07M.2.hl.TZ0.4a: The function \(f\) is defined by \(f(x) = \frac{{{{\rm{e}}^x} + {{\rm{e}}^{ - x}}}}{2}\) .  ...
• 07M.2.hl.TZ0.4b: Use the integral test to determine whether the...
• 12M.1.hl.TZ0.3a: By evaluating successive derivatives at \(x = 0\) , find the Maclaurin series for \(\ln \cos x\)...
• 12M.1.hl.TZ0.3b: Consider \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \cos x}}{{{x^n}}}\) , where...
• 12M.2.hl.TZ0.3a: (i)     Show that...
• 12M.2.hl.TZ0.3b: Consider the differential equation...
• SPNone.1.hl.TZ0.8b: Show that \(S > 0.4\) .
• SPNone.1.hl.TZ0.1: Using l’Hôpital’s Rule, determine the value...
• SPNone.1.hl.TZ0.8a: Show that the series is conditionally convergent but not absolutely convergent.
• SPNone.1.hl.TZ0.11a: Show that \(f''(x) = - 2{{\rm{e}}^x}\sin x\) .
• SPNone.1.hl.TZ0.11b: Determine the Maclaurin series for \(f(x)\) up to and including the term in \({x^4}\) .
• SPNone.1.hl.TZ0.11c: By differentiating your series, determine the Maclaurin series for \({{\rm{e}}^x}\sin x\) up to...
• SPNone.2.hl.TZ0.5a: Use Euler’s method with a step length of \(0.1\) to find an approximate value for \(y\) when...
• SPNone.2.hl.TZ0.5b: (i)     By differentiating the above differential equation, obtain an expression involving...
• SPNone.2.hl.TZ0.5c: (i)     Show that \(\sec x + \tan x\) is an integrating factor for solving this differential...
• 14M.1.hl.TZ0.2: Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^3} - {x^3}\) for...
• 14M.1.hl.TZ0.5: (a)     Assuming the Maclaurin series for \({{\text{e}}^x}\), determine the first three non-zero...
• 14M.1.hl.TZ0.12: Consider the infinite series...
• 14M.2.hl.TZ0.5: Consider the differential equation...
• 14M.2.hl.TZ0.8: (a)     (i)     Using l’Hôpital’s rule, show...
• 15M.1.hl.TZ0.1: Use l’Hôpital’s rule to find \(\mathop {\lim }\limits_{x \to 0} (\csc x - \cot x)\).
• 15M.1.hl.TZ0.8a: Differentiate the expression \({x^2}\tan y\) with respect to \(x\), where \(y\) is a function of...