DP Mathematics HL Questionbank

• Syllabus

Topic 9 - Option: Calculus

Sub sections and their related questions

9.1

• 12M.3ca.hl.TZ0.4a: Show that the sequence converges to a limit L , the value of which should be stated.
• 12M.3ca.hl.TZ0.4b: Find the least value of the integer N such that \(\left| {{u_n} - L} \right| < \varepsilon...
• 12M.3ca.hl.TZ0.4c: For each of the sequences...
• 13N.3ca.hl.TZ0.2: The general term of a sequence \(\{ {a_n}\} \) is given by the formula...

9.2

• 12M.3ca.hl.TZ0.4d: Prove that the series \(\sum\limits_{n = 1}^\infty  {({u_n} - L)} \) diverges.
• 12M.3ca.hl.TZ0.5b: Show that the series \(\sum\limits_{r = 2}^\infty {\frac{{{{( - 1)}^r}}}{{r\ln r}}} \) is...
• 12N.3ca.hl.TZ0.3b: Use the integral test to prove that \(\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}} \) converges.
• 12N.3ca.hl.TZ0.4a: Use the limit comparison test to prove that...
• 08M.3ca.hl.TZ1.1: Determine whether the series \(\sum\limits_{n = 1}^\infty  {\frac{{{n^{10}}}}{{{{10}^n}}}} \) is...
• 08M.3ca.hl.TZ2.5a: Find the radius of convergence of the series...
• 08M.3ca.hl.TZ2.5b: Determine whether the series...
• 08N.3ca.hl.TZ0.2a: (i)     Show that \(\int_1^\infty  {\frac{1}{{x(x + p)}}{\text{d}}x,{\text{ }}p \ne 0} \) is...
• 08N.3ca.hl.TZ0.2b: Determine, for each of the following series, whether it is convergent or divergent. (i)    ...
• 08N.3ca.hl.TZ0.3: The function \(f(x) = \frac{{1 + ax}}{{1 + bx}}\) can be expanded as a power series in x, within...
• 11M.3ca.hl.TZ0.5a: Find the set of values of x for which the series is convergent.
• 11M.3ca.hl.TZ0.5b: (i)     Show, by comparison with an appropriate geometric series,...
• 11M.3ca.hl.TZ0.5c: (i)     Write down the first three terms of the Maclaurin series for \(1 - {{\text{e}}^{ - x}}\)...
• 11M.3ca.hl.TZ0.5d: Letting n = 1000, use the results in parts (b) and (c) to calculate the value of e correct to as...
• 09M.3ca.hl.TZ0.3a: Determine whether the series \(\sum\limits_{n = 1}^\infty {\sin \frac{1}{n}} \) is convergent or...
• 09M.3ca.hl.TZ0.3b: Show that the series \(\sum\limits_{n = 2}^\infty {\frac{1}{{n{{(\ln n)}^2}}}} \) is convergent.
• 09N.3ca.hl.TZ0.5a: Find the radius of convergence of the infinite...
• 09N.3ca.hl.TZ0.5b: Determine whether the series...
• SPNone.3ca.hl.TZ0.5a: Find the radius of convergence.
• SPNone.3ca.hl.TZ0.5b: Find the interval of convergence.
• 10M.3ca.hl.TZ0.5a: Consider the power series \(\sum\limits_{k = 1}^\infty {k{{\left( {\frac{x}{2}} \right)}^k}}...
• 10M.3ca.hl.TZ0.5b: Consider the infinite series...
• 10N.3ca.hl.TZ0.2: Determine whether or not the following series converge. (a)    ...
• 10N.3ca.hl.TZ0.5: Consider the infinite...
• 13M.3ca.hl.TZ0.3a: Find the radius of convergence.
• 13M.3ca.hl.TZ0.3b: Find the interval of convergence.
• 13M.3ca.hl.TZ0.3c: Given that x = – 0.1, find the sum of the series correct to three significant figures.
• 11N.3ca.hl.TZ0.2b: Hence use the comparison test to determine whether the series...
• 11N.3ca.hl.TZ0.3a: Find the radius of convergence of the series.
• 11N.3ca.hl.TZ0.3b: Hence deduce the interval of convergence.
• 11N.3ca.hl.TZ0.4a: Using the integral test, show that \(\sum\limits_{n = 1}^\infty  {\frac{1}{{4{n^2} + 1}}} \) is...
• 14M.3ca.hl.TZ0.3: Each term of the power series...
• 13N.3ca.hl.TZ0.1a: Consider the infinite series \(\sum\limits_{n = 1}^\infty  {\frac{2}{{{n^2} + 3n}}} \). Use a...
• 13N.3ca.hl.TZ0.5: A function \(f\) is defined in the interval \(\left] { - k,{\text{ }}k} \right[\), where...
• 15M.3ca.hl.TZ0.3a: Show that the series \(\sum\limits_{n = 2}^\infty  {\frac{1}{{{n^2}\ln n}}} \) converges.
• 15M.3ca.hl.TZ0.3c: (i)     State why the integral test can be used to determine the convergence or divergence of...
• 15N.3ca.hl.TZ0.3b: Hence use the comparison test to prove that the series...
• 15N.3ca.hl.TZ0.4c: Show that...
• 16M.3ca.hl.TZ0.5a: Explain why the series is alternating.
• 16M.3ca.hl.TZ0.5b: (i)     Use the substitution \(T = t - \pi \) in the expression for \({u_{n + 1}}\) to show that...
• 16M.3ca.hl.TZ0.5c: Show that \(S < 1.65\).
• 16N.3ca.hl.TZ0.3b: (i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\). (ii) Show...
• 16N.3ca.hl.TZ0.3c: Using a suitable test, determine whether this series converges or diverges.
• 17M.3ca.hl.TZ0.3: Use the integral test to determine whether the infinite series...
• 17N.3ca.hl.TZ0.3a: Use the limit comparison test to show that the series...
• 17N.3ca.hl.TZ0.3b: Find the interval of convergence for \(S\).
• 18M.3ca.hl.TZ0.1a: Given that \(n > {\text{ln}}\,n\) for \(n > 0\), use the comparison test to show that the...
• 18M.3ca.hl.TZ0.1b: Find the interval of convergence...

9.3

• SPNone.3ca.hl.TZ0.4a: Prove that f is continuous but not differentiable at the point (0, 0) .
• 14M.3ca.hl.TZ0.4a: Find the exact values of \(a\) and \(b\) if \(f\) is continuous and differentiable at \(x = 1\).
• 17N.3ca.hl.TZ0.1: The function \(f\) is defined...
• 18M.3ca.hl.TZ0.2: The function \(f\) is defined...

9.4

• 12M.3ca.hl.TZ0.5a: Find the set of values of k for which the improper integral...
• 12N.3ca.hl.TZ0.3a: Prove that \(\mathop {\lim }\limits_{H \to \infty } \int_a^H {\frac{1}{{{x^2}}}{\text{d}}x} \)...
• 12N.3ca.hl.TZ0.3c: Let \(\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}}  = L\) . The diagram below shows the graph...
• 12N.3ca.hl.TZ0.3d: Hence show that...
• 12N.3ca.hl.TZ0.3e: You are given that \(L = \frac{{{\pi ^2}}}{6}\). By taking k = 4 , use the upper bound and lower...
• 08M.3ca.hl.TZ1.2: (a)     Using l’Hopital’s Rule, show that...
• 08M.3ca.hl.TZ1.4: (a)     Using the diagram, show that...
• 08M.3ca.hl.TZ2.2: Find the exact value of \(\int_0^\infty  {\frac{{{\text{d}}x}}{{(x + 2)(2x + 1)}}} \).
• 13M.3ca.hl.TZ0.5a: Figure 1 shows part of the graph of \(y = \frac{1}{x}\) together with line segments parallel to...
• 13M.3ca.hl.TZ0.5b: An improved upper bound can be found by considering Figure 2 which again shows part of the graph...
• 11N.3ca.hl.TZ0.4b: (i)     Show, by means of a diagram, that...
• 14M.3ca.hl.TZ0.1d: Find the value of the improper integral...
• 15N.3ca.hl.TZ0.4b: Use the inequality in part (a) to find a lower and upper bound for \(\pi \).
• 16M.3ca.hl.TZ0.2a: Write down \(f'(x)\).
• 16M.3ca.hl.TZ0.2b: By differentiating \(f({x^2})\), obtain an expression for the derivative of...
• 16M.3ca.hl.TZ0.2c: Hence obtain an expression for the derivative of...
• 17M.3ca.hl.TZ0.5a: By drawing a diagram and considering the area of a suitable region under the curve, show that for...
• 17M.3ca.hl.TZ0.5b.i: Hence, given that \(n\) is a positive integer greater than one, show...
• 17M.3ca.hl.TZ0.5b.ii: Hence, given that \(n\) is a positive integer greater than one, show...
• 17M.3ca.hl.TZ0.5c.i: Hence, given that \(n\) is a positive integer greater than one, show that \({U_n} > 0\);
• 17M.3ca.hl.TZ0.5c.ii: Hence, given that \(n\) is a positive integer greater than one, show...
• 17M.3ca.hl.TZ0.5d: Explain why these two results prove that \(\{ {U_n}\} \) is a convergent sequence.
• 18M.3ca.hl.TZ0.3a: Find the value of \(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \).
• 18M.3ca.hl.TZ0.3b: Illustrate graphically the...
• 18M.3ca.hl.TZ0.3c: Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).
• 18M.3ca.hl.TZ0.3d: Find an upper bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

9.5

• 12M.2.hl.TZ2.12a: Find an expression for v in terms of t .
• 12M.3ca.hl.TZ0.2a: Use Euler’s method, with a step length of 0.1, to find an approximate value of y when x = 0.5.
• 12M.3ca.hl.TZ0.2c: (i)     Solve the differential equation. (ii)     Find the value of a for which...
• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 12N.3ca.hl.TZ0.1a: Solve this differential equation by separating the variables, giving your answer in the form y =...
• 12N.3ca.hl.TZ0.1b: Solve the same differential equation by using the standard homogeneous substitution y = vx .
• 12N.3ca.hl.TZ0.1c: Solve the same differential equation by the use of an integrating factor.
• 12N.3ca.hl.TZ0.1d: If y = 20 when x = 2 , find y when x = 5 .
• 12N.3ca.hl.TZ0.2a: Use Euler’s method to find an approximation for the value of c , using a step length of h = 0.1 ....
• 12N.3ca.hl.TZ0.2b: You are told that if Euler’s method is used with h = 0.05 then \(c \simeq 2.7921\) , if it is...
• 12N.3ca.hl.TZ0.2c: Draw, by eye, the straight line that best fits these four points, using a ruler.
• 12N.3ca.hl.TZ0.2d: Use your graph to give the best possible estimate for c , giving your answer to three decimal...
• 08M.3ca.hl.TZ1.3: (a)     Find an integrating factor for this differential equation. (b)     Solve the...
• 08M.3ca.hl.TZ2.3: (a)     (i)     Use Euler’s method to get an approximate value of y when x = 1.3 , taking steps...
• 08N.3ca.hl.TZ0.1: (a)     Show that the solution of the homogeneous differential...
• 08N.3ca.hl.TZ0.4: (a)     Show that the solution of the differential...
• 08M.1.hl.TZ1.13: A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its...
• 08N.2.hl.TZ0.9: The population of mosquitoes in a specific area around a lake is controlled by pesticide. The...
• 11M.2.hl.TZ2.13B: (a)     Using integration by parts, show that...
• 11M.3ca.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of y when x = 0.4.
• 11M.3ca.hl.TZ0.2b: Write down, giving a reason, whether your approximate value for y is greater than or less than...
• 11M.3ca.hl.TZ0.3: Solve the differential...
• 09M.3ca.hl.TZ0.2: The variables x and y are related by \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - y\tan x = \cos x\)...
• 09M.3ca.hl.TZ0.4: Consider the differential equation...
• 09M.1.hl.TZ1.13Part B: Let f be a function with domain \(\mathbb{R}\) that satisfies the...
• 09N.1.hl.TZ0.8: A certain population can be modelled by the differential equation...
• 09N.3ca.hl.TZ0.1: Solve the differential...
• SPNone.3ca.hl.TZ0.2a: Show that this is a homogeneous differential equation.
• SPNone.3ca.hl.TZ0.2b: Find the general solution, giving your answer in the form \(y = f(x)\) .
• SPNone.3ca.hl.TZ0.3a: By finding the values of successive derivatives when x = 0 , find the Maclaurin series for y as...
• SPNone.3ca.hl.TZ0.3b: (i)     Differentiate the function \({{\text{e}}^x}(\sin x + \cos x)\) and hence show...
• 10M.3ca.hl.TZ0.1: Given that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2{y^2} = {{\text{e}}^x}\) and y = 1 when x =...
• 10M.3ca.hl.TZ0.3: Solve the differential...
• 10N.1.hl.TZ0.8: Find y in terms of x, given that...
• 10N.3ca.hl.TZ0.4: Solve the differential...
• 13M.3ca.hl.TZ0.2a: Use Euler’s method with a step length of 0.1 to find an approximation to the value of y when x =...
• 13M.3ca.hl.TZ0.2b: (i)     Show that the integrating factor for solving the differential equation is \(\sec...
• 13M.2.hl.TZ2.10: The acceleration of a car is \(\frac{1}{{40}}(60 - v){\text{ m}}{{\text{s}}^{ - 2}}\), when its...
• 13M.2.hl.TZ2.12a: (i)     Show that the function \(y = \cos x + \sin x\) satisfies the differential equation. (ii)...
• 13M.2.hl.TZ2.12b: A different solution of the differential equation, satisfying y = 2 when \(x = \frac{\pi }{4}\),...
• 11N.1.hl.TZ0.13b: Find \(f(x)\).
• 11N.1.hl.TZ0.13c: Determine the largest possible domain of f.
• 11N.1.hl.TZ0.13d: Show that the equation \(f(x) = f'(x)\) has no solution.
• 11N.3ca.hl.TZ0.6: The real and imaginary parts of a complex number \(x + {\text{i}}y\) are related by the...
• 11M.2.hl.TZ1.14c: If the glass is filled completely, how long will it take for all the water to evaporate?
• 09M.2.hl.TZ1.8: (a)     Solve the differential equation...
• 09M.2.hl.TZ2.6: The acceleration in ms−2 of a particle moving in a straight line at time \(t\) seconds,...
• 14M.3ca.hl.TZ0.2b: Consider the differential...
• 13N.3ca.hl.TZ0.3: Consider the differential equation...
• 14N.3ca.hl.TZ0.2a: Use an integrating factor to show that the general solution for...
• 14N.3ca.hl.TZ0.2b: Given that \(w(t)\) is continuous, find the value of \(c\).
• 14N.3ca.hl.TZ0.2c: Write down (i)     the weight of the dog when bought from the pet shop; (ii)     an upper bound...
• 14N.3ca.hl.TZ0.3a: Sketch, on one diagram, the four isoclines corresponding to \(f(x,{\text{ }}y) = k\) where \(k\)...
• 14N.3ca.hl.TZ0.3b: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 14N.3ca.hl.TZ0.3c: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 14N.3ca.hl.TZ0.3d: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 15M.3ca.hl.TZ0.2a: Show that \(y = \frac{1}{x}\int {f(x){\text{d}}x} \) is a solution of the differential...
• 15M.3ca.hl.TZ0.2b: Hence solve...
• 15N.3ca.hl.TZ0.5a: Show that the tangent to the curve \(y = f(x)\) at the point \((1,{\text{ }}0)\) is normal to the...
• 15N.3ca.hl.TZ0.5b: Find \(g(x)\).
• 15N.3ca.hl.TZ0.5c: Use Euler’s method with steps of \(0.2\) to estimate \(f(2)\) to \(5\) decimal places.
• 15N.3ca.hl.TZ0.5d: Explain why \(y = f(x)\) cannot cross the isocline \(x - {y^2} = 0\), for \(x > 1\).
• 15N.3ca.hl.TZ0.5e: (i)     Sketch the isoclines \(x - {y^2} =  - 2,{\text{ }}0,{\text{ }}1\). (ii)     On the same...
• 16M.3ca.hl.TZ0.4a: Show that putting \(z = {y^2}\) transforms the differential equation into...
• 16M.3ca.hl.TZ0.4b: By solving this differential equation in \(z\), obtain an expression for \(y\) in terms of \(x\).
• 16N.3ca.hl.TZ0.1a: Show that \(1 + {x^2}\) is an integrating factor for this differential equation.
• 16N.3ca.hl.TZ0.1b: Hence solve this differential equation. Give the answer in the form \(y = f(x)\).
• 17M.3ca.hl.TZ0.4a: Consider the differential...
• 17M.3ca.hl.TZ0.4b: Hence, or otherwise, solve the differential...
• 17N.3ca.hl.TZ0.2a: Show that \(\sqrt {{x^2} + 1} \) is an integrating factor for this differential equation.
• 17N.3ca.hl.TZ0.2b: Solve the differential equation giving your answer in the form \(y = f(x)\).
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

9.6

• 12M.3ca.hl.TZ0.2b: (i)     Show that...
• 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...
• 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...
• 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...
• 14N.3ca.hl.TZ0.4b: Hence show that an expansion of \(\arctan x\) is...
• 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 \(\frac{{16}}{{3\sqrt 3 }} < \pi  < \frac{6}{{\sqrt 3 }}\).
• 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...
• 15N.3ca.hl.TZ0.2b: By further differentiation of the result in part (a) , find the Maclaurin expansion of \(f(x)\),...
• 15N.3ca.hl.TZ0.4d: Hence show that...
• 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...
• 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 \(h > 0\) \(1 - \cos (h) \leqslant \frac{{{h^2}}}{2}\).
• 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...
• 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.TZ0.2b: Hence, by comparing your two series, determine the values of \({a_1}\), \({a_3}\) and \({a_5}\).
• 17N.3ca.hl.TZ0.4a: For \(a = 0\) and \(b = 5\pi \), use the mean value theorem to find all possible values of \(c\)...
• 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.TZ0.5a: Show that \(f’(0) = p\).
• 17N.3ca.hl.TZ0.5b: Show that \({f^{(n + 2)}}(0) = ({n^2} - {p^2}){f^{(n)}}(0)\).
• 17N.3ca.hl.TZ0.5c: For \(p \in \mathbb{R}\backslash \{  \pm 1,{\text{ }} \pm 3\} \), show that the Maclaurin series...
• 17N.3ca.hl.TZ0.5d: Hence or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\sin (p\arcsin x)}}{x}\).
• 17N.3ca.hl.TZ0.5e: If \(p\) is an odd integer, prove that the Maclaurin series for \(f(x)\) is a polynomial of...
• 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...

9.7

• 12M.3ca.hl.TZ0.1: Use L’Hôpital’s Rule to find...
• 08M.3ca.hl.TZ1.2: (a)     Using l’Hopital’s Rule, show that...
• 08M.3ca.hl.TZ2.1a: Find the value of...
• 08M.3ca.hl.TZ2.1b: By using the series expansions for \({{\text{e}}^{{x^2}}}\) and cos x evaluate...
• 08N.3ca.hl.TZ0.4: (a)     Show that the solution of the differential...
• 11M.3ca.hl.TZ0.1b: Hence, or otherwise, determine the value of...
• 09M.3ca.hl.TZ0.1a: Find \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{{x + {x^2}}}\) ;
• 09M.3ca.hl.TZ0.1b: Find...
• 09N.3ca.hl.TZ0.2: The function f is defined by \(f(x) = {{\text{e}}^{({{\text{e}}^x} - 1)}}\) . (a)     Assuming...
• SPNone.3ca.hl.TZ0.1c: Determine the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln (1 + \sin x) - x}}{{{x^2}}}\).
• 10M.3ca.hl.TZ0.4: (a)     Using the Maclaurin series for \({(1 + x)^n}\), write down and simplify the Maclaurin...
• 10N.3ca.hl.TZ0.1: Find \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 - \cos {x^6}}}{{{x^{12}}}}} \right)\).
• 13M.3ca.hl.TZ0.1b: Hence, or otherwise, find the value of...
• 11N.3ca.hl.TZ0.1: Find...
• 14M.3ca.hl.TZ0.1c: Hence find the value of \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 - f(x)}}{{{x^2}}}\).
• 13N.3ca.hl.TZ0.4a: Using the result \(\mathop {{\text{lim}}}\limits_{t \to 0} \frac{{\sin t}}{t} = 1\), or...
• 16N.3ca.hl.TZ0.3a: Using l’Hôpital’s rule, find...
• 17M.3ca.hl.TZ0.1: Use l’Hôpital’s rule to determine the value...