DP Mathematics HL Questionbank
9.6
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[N/A]Directly related questions
- 18M.3ca.hl.TZ0.4d: Use this series approximation for \(f\left( x \right)\) with \(x = \frac{1}{2}\) to find an...
- 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.TZ0.4b: By differentiating the above equation twice, show...
- 18M.3ca.hl.TZ0.4a: Show that \(f'\left( 0 \right) = 0\).
- 16M.3ca.hl.TZ0.3b: Hence show that \(\ln (1.2)\) lies between \(\frac{1}{m}\) and \(\frac{1}{n}\), where \(m\),...
- 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.TZ0.1c: (i) Use the Lagrange form of the error term to find an upper bound for the absolute value of...
- 16M.3ca.hl.TZ0.1b: Hence, or otherwise, determine the exact value of...
- 16M.3ca.hl.TZ0.1a: By finding a suitable number of derivatives of \(f\), determine the Maclaurin series for \(f(x)\)...
- 16N.3ca.hl.TZ0.4c: Hence show that, for \(h > 0\) \(1 - \cos (h) \leqslant \frac{{{h^2}}}{2}\).
- 16N.3ca.hl.TZ0.4b: (i) Find \(g(0)\). (ii) Find \(g(h)\). (iii) Apply the mean value theorem to the...
- 16N.3ca.hl.TZ0.4a: State the mean value theorem for a function that is continuous on the closed interval...
- 16N.3ca.hl.TZ0.2c: By applying the ratio test, find the radius of convergence for this Maclaurin series.
- 16N.3ca.hl.TZ0.2b: Deduce that, for \(n \geqslant 2\), the coefficient of \({x^n}\) in this series is...
- 16N.3ca.hl.TZ0.2a: By successive differentiation find the first four non-zero terms in the Maclaurin series for...
- 17N.3ca.hl.TZ0.5e: If \(p\) is an odd integer, prove that the Maclaurin series for \(f(x)\) is a polynomial of...
- 17N.3ca.hl.TZ0.5d: Hence or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin (p\arcsin x)}}{x}\).
- 17N.3ca.hl.TZ0.5c: For \(p \in \mathbb{R}\backslash \{ \pm 1,{\text{ }} \pm 3\} \), show that the Maclaurin series...
- 17N.3ca.hl.TZ0.5b: Show that \({f^{(n + 2)}}(0) = ({n^2} - {p^2}){f^{(n)}}(0)\).
- 17N.3ca.hl.TZ0.5a: Show that \(f’(0) = p\).
- 17N.3ca.hl.TZ0.4b: Sketch the graph of \(y = g(x)\) on the interval \([0,{\text{ }}5\pi ]\) and hence illustrate the...
- 17N.3ca.hl.TZ0.4a: For \(a = 0\) and \(b = 5\pi \), use the mean value theorem to find all possible values of \(c\)...
- 17M.3ca.hl.TZ0.2b: Hence, by comparing your two series, determine the values of \({a_1}\), \({a_3}\) and \({a_5}\).
- 17M.3ca.hl.TZ0.2a.ii: Find series for \({\sec ^2}x\), in terms of \({a_1}\), \({a_3}\) and \({a_5}\), up to and...
- 17M.3ca.hl.TZ0.2a.i: Find series for \({\sec ^2}x\), in terms of \({a_1}\), \({a_3}\) and \({a_5}\), up to and...
- 15N.3ca.hl.TZ0.4d: Hence show that...
- 15N.3ca.hl.TZ0.2b: By further differentiation of the result in part (a) , find the Maclaurin expansion of \(f(x)\),...
- 12M.3ca.hl.TZ0.2b: (i) Show that...
- 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...
- 08N.3ca.hl.TZ0.1: (a) Show that the solution of the homogeneous differential...
- 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)...
- SPNone.3ca.hl.TZ0.3a: By finding the values of successive derivatives when x = 0 , find the Maclaurin series for y as...
- 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...
- 13M.3ca.hl.TZ0.1a: Find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2}\) and \({a_3}\).
- 11N.3ca.hl.TZ0.5b: Hence, by repeated differentiation of the above differential equation, find the Maclaurin series...
- 12N.3ca.hl.TZ0.4c: Using the Maclaurin series for \(\ln (1 + x)\) , show that the Maclaurin series for...
- 14M.3ca.hl.TZ0.1b: Find the first three non-zero terms in the Maclaurin expansion of \(f(x)\).
- 14M.3ca.hl.TZ0.4b: (i) Use Rolle’s theorem, applied to \(f\), to prove that...
- 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...
- 15M.3ca.hl.TZ0.1: The function \(f\) is defined by \(f(x) = {{\text{e}}^{ - x}}\cos x + x - 1\). By finding a...
- 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...
- 14N.3ca.hl.TZ0.4c: \(f\) is a continuous function defined on \([a,{\text{ }}b]\) and differentiable on...
- 14N.3ca.hl.TZ0.4f: Hence show that \(\frac{{16}}{{3\sqrt 3 }} < \pi < \frac{6}{{\sqrt 3 }}\).
- 14N.3ca.hl.TZ0.4b: Hence show that an expansion of \(\arctan x\) is...
- 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...
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...
