DP Further Mathematics HL Questionbank

• Syllabus

5.6

Sub sections and their related questions

Rolle’s theorem.

• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

Mean value theorem.

• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

Taylor polynomials; the Lagrange form of the error term.

• 15M.2.hl.TZ0.8a: Using a Taylor series, find a quadratic approximation for \(f(x) = \sin x\) centred about...
• 15M.2.hl.TZ0.8b: When using this approximation to find angles between \(130^\circ\) and \(140^\circ\), find the...
• 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.TZ0.8d: Explain briefly why the same maximum value of error term occurs for \(g(x) = \cos x\) centred...
• 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.TZ0.7b: (i)     Use these results to show that the Maclaurin series for the function \({f_5}(x)\) up to...
• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

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

• 11M.2.hl.TZ0.5c: (i)     Find the Maclaurin series for \(\ln (1 + \sin x)\) up to and including the term in...
• 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.TZ0.1A.b: (i)      Use your answer to (a) to find an approximate expression for the cumulative distributive...
• 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.TZ0.2A.d: By combining your two series, show that...
• 07M.2.hl.TZ0.4a: The function \(f\) is defined by \(f(x) = \frac{{{{\rm{e}}^x} + {{\rm{e}}^{ - x}}}}{2}\) .  ...
• 12M.1.hl.TZ0.3a: By evaluating successive derivatives at \(x = 0\) , find the Maclaurin series for \(\ln \cos x\)...
• 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...
• 14M.1.hl.TZ0.5: (a)     Assuming the Maclaurin series for \({{\text{e}}^x}\), determine the first three non-zero...
• 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.TZ0.7b: (i)     Use these results to show that the Maclaurin series for the function \({f_5}(x)\) up to...
• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

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

• 15M.2.hl.TZ0.1d: (i)     Find the first three non-zero terms of the Maclaurin series for \(y\). (ii)     Hence...
• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).

Taylor series developed from differential equations.

• SPNone.2.hl.TZ0.5b: (i)     By differentiating the above differential equation, obtain an expression involving...
• 14M.2.hl.TZ0.5: Consider the differential equation...
• 15M.2.hl.TZ0.1d: (i)     Find the first three non-zero terms of the Maclaurin series for \(y\). (ii)     Hence...
• 18M.2.hl.TZ0.2b.iii: Find the Maclaurin expansion for \(y\) up to and including the term in \({{x^3}}\).