IB Questionbank

Processing math: 26%

Updates to Questionbank

User interface language: English | Español

Date	November 2018	Marks available	7	Reference code	18N.2.AHL.TZ0.H_2
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	H_2	Adapted from	N/A

Question

A function $f$ satisfies the conditions $f\left( 0 \right) = - 4$ , $f\left( 1 \right) = 0$ and its second derivative is $f''\left( x \right) = 15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}$ , $x$ ≥ 0.

Find $f\left( x \right)$ .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$f'\left( x \right) = \int {\left( {15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}} \right)} \,{\text{d}}x = 10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}}\left( { + c} \right)$ (M1)A1A1

Note: A1 for first term, A1 for second term. Withhold one A1 if extra terms are seen.

$f\left( x \right) = \int {\left( {10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}} + c} \right)} \,{\text{d}}x = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + cx + d$ A1

Note: Allow FT from incorrect $f'\left( x \right)$ if it is of the form $f'\left( x \right) = A{x^{\frac{3}{2}}} + \frac{B}{{x + 1}} + c$ .

Accept ${\text{ln}}\left| {x + 1} \right|$ .

attempt to use at least one boundary condition in their $f\left( x \right)$ (M1)

$x = 0$ , $y = - 4$

⇒ $d = - 4$ A1

$x = 1$ , $y = 0$

⇒ $0 = 4 - {\text{ln}}\,2 + c - 4$

⇒ $c = {\text{ln}}\,2\left( { = 0.693} \right)$ A1

$f\left( x \right) = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + x\,{\text{ln}}\,2 - 4$

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.11—Indefinite integration, reverse chain, by substitution

Show 34 related questions

22M.1.AHL.TZ1.14a.ii:
Find $\int {(\frac{1}{(x + 2)} + 1)}^{2} d x$ .
18M.1.SL.TZ1.S_5a:
Find $\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x}$ .
17M.1.SL.TZ2.S_5:
Let $f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}$ . Given that $f(0) = 1$ , find $f(x)$ .
22M.1.AHL.TZ1.14a.i:
Expand ${(\frac{1}{u} + 1)}^{2}$ .
18N.1.SL.TZ0.S_6:
Let $f\left( x \right) = \frac{{6 - 2x}}{{\sqrt {16 + 6x - {x^2}} }}$ . The following diagram shows part of the graph of $f$ .

The region R is enclosed by the graph of $f$ , the $x$ -axis, and the $y$ -axis. Find the area of R.
19M.1.SL.TZ2.S_10c:
Write down an expression for the area of $R$ .
19M.1.AHL.TZ1.H_8d:
Sketch the curve $y = f\left( x \right)$ , 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.
19M.1.SL.TZ2.S_10d:
Hence find the exact area of $R$ .
20N.2.SL.TZ0.S_10b:
Find the least value of $n$ .
18M.1.AHL.TZ2.H_6a.i:
Find $f'\left( x \right)$ .
18M.1.AHL.TZ2.H_6a.ii:
Find $g'\left( x \right)$ .
19M.1.SL.TZ2.S_10b:
Hence find $\int {\left( {3{x^2} + 1} \right)\sqrt {{x^3} + x} } \,{\text{d}}x$ .
19M.1.AHL.TZ1.H_8a:
Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .
18M.1.AHL.TZ1.H_4b:
$\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$ .
16N.1.SL.TZ0.S_6:
Let $f'(x) = {\sin ^3}(2x)\cos (2x)$ . Find $f(x)$ , given that $f\left( {\frac{\pi }{4}} \right) = 1$ .
20N.2.SL.TZ0.S_10a:
Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .
17M.1.SL.TZ1.S_5a:
Find $\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x}$ .
EXN.1.AHL.TZ0.10a.ii:
State in context what this value represents.
EXN.1.AHL.TZ0.10a.i:
Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .
18M.1.SL.TZ1.S_5b:
Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.
18M.1.SL.TZ2.S_2a:
Find $\int {\left( {6{x^2} - 3x} \right){\text{d}}x}$ .
18M.1.AHL.TZ2.H_6b:
Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .
18M.1.AHL.TZ1.H_4a:
$\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$ .
19M.1.SL.TZ2.S_10a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
20N.2.SL.TZ0.S_10c:
Find $\int \frac{6 x}{x^{2} + 4} d x$ .
19M.1.AHL.TZ1.H_8b:
find the value of $f\left( 1 \right)$ .
19M.1.SL.TZ1.S_5:
The derivative of a function $f$ is given by $f'(x) = 2{{\text{e}}^{ - 3x}}$ . The graph of $f$ passes through $\left( {\frac{1}{3}{\text{,}}\,\,5} \right)$ .

Find $f(x)$ .
18M.1.SL.TZ2.S_2b:
Find the area of the region enclosed by the graph of $f$ , the x-axis and the lines x = 1 and x = 2 .
20N.1.SL.TZ0.S_6:
The graph of a function $f$ passes through the point $(\ln 4, 20)$ .

Given that $f' (x) = 6 e^{2 x}$ , find $f (x)$ .
20N.2.SL.TZ0.S_10d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ . The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
17M.1.SL.TZ1.S_5b:
Find $f(x)$ , given that $f’(x) = x{{\text{e}}^{{x^2} - 1}}$ and $f( - 1) = 3$ .
EXN.1.AHL.TZ0.10c:
Determine $\int_{0}^{365} P (t) d t$ and state what it represents.
EXN.1.AHL.TZ0.10b:
Find an expression for $P$ in terms of $t$ .

Hide related questions

Topic 5—Calculus