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...