LFM Pure and Mechanics

tmua 2019 Q14 1 marks View

Consider the following statements about the polynomial $\mathrm { p } ( x )$, where $a < b$ :
I $\quad \mathrm { p } ( a ) \leq \mathrm { p } ( b )$
II $\quad \mathrm { p } ^ { \prime } ( a ) \leq \mathrm { p } ^ { \prime } ( b )$
III $\mathrm { p } ^ { \prime \prime } ( a ) \leq \mathrm { p } ^ { \prime \prime } ( b )$
Which of these statements is a necessary condition for $\mathrm { p } ( x )$ to be increasing for $a \leq x \leq b$ ?

tmua 2019 Q18 1 marks Geometric or applied optimisation problem View

Find the shortest distance between the curve $y = x^2 + 4$ and the line $y = 2x - 2$.

tmua 2020 Q8 1 marks Determine parameters from given extremum conditions View

The function f is defined for all real $x$ as
$$\mathrm{f}(x) = (p-x)(x+2)$$
Find the complete set of values of $p$ for which the maximum value of $\mathrm{f}(x)$ is less than 4.
A $-2 - 4\sqrt{2} < p < -2 + 4\sqrt{2}$
B $-2 - 2\sqrt{2} < p < -2 + 2\sqrt{2}$
C $-2\sqrt{5} < p < 2\sqrt{5}$
D $-6 < p < 2$
E $-4 < p < 0$
F $-2 < p < 2$

tmua 2021 Q11 1 marks Determine intervals of increase/decrease or monotonicity conditions View

The function f is given by
$$\mathrm { f } ( x ) = x ^ { \frac { 1 } { 7 } } \left( x ^ { 2 } - x + 1 \right)$$
Find the fraction of the interval $0 < x < 1$ for which $\mathrm { f } ( x )$ is decreasing.
A $\frac { 2 } { 15 }$
B $\frac { 1 } { 5 }$
C $\frac { 1 } { 3 }$
D $\frac { 1 } { 2 }$
E $\frac { 2 } { 3 }$
F $\frac { 4 } { 5 }$
G $\frac { 13 } { 15 }$

tmua 2021 Q12 1 marks Determine parameters from given extremum conditions View

The minimum value of the function $x ^ { 4 } - p ^ { 2 } x ^ { 2 }$ is - 9 $p$ is a real number.
Find the minimum value of the function $x ^ { 2 } - p x + 6$
A - 3 B $6 - \frac { 3 \sqrt { 2 } } { 2 }$ C $\frac { 3 } { 2 }$ D 3 E $\frac { 9 } { 2 }$ F $6 + \frac { 3 \sqrt { 2 } } { 2 }$

tmua 2022 Q1 1 marks Find critical points and classify extrema of a given function View

Determine the number of stationary points on the curve with equation
$$y = 3 x ^ { 4 } + 4 x ^ { 3 } + 6 x ^ { 2 } - 5$$
A 0
B 1
C 2
D 3
E 4

tmua 2022 Q13 1 marks Find absolute extrema on a closed interval or domain View

Given that
$$\left( a ^ { 3 } + \frac { 2 } { b ^ { 3 } } \right) \left( \frac { 2 } { a ^ { 3 } } - b ^ { 3 } \right) = \sqrt { 2 }$$
where $a$ and $b$ are real numbers, what is the least value of $a b$ ?

tmua 2023 Q5 1 marks Geometric or applied optimisation problem View

The following shape has two lines of reflectional symmetry.
$M N O P$ is a square of perimeter 40 cm .
The vertices of rectangle $R S T U$ lie on the edge of square $M N O P$.
$M R$ has length $x \mathrm {~cm}$.
What is the largest possible value of $x$ such that $R S T U$ has area $20 \mathrm {~cm} ^ { 2 }$ ?

tmua 2023 Q14 1 marks Count or characterize roots using extremum values View

The function
$$f ( x ) = \frac { 2 } { 3 } x ^ { 3 } + 2 m x ^ { 2 } + n , \quad m > 0$$
has three distinct real roots.
What is the complete range of possible values of $n$, in terms of $m$ ?

tmua 2023 Q20 1 marks Find absolute extrema on a closed interval or domain View

The diagram shows the graph of $y = f ( x )$
The function $f$ attains its maximum value of 2 at $x = 1$, and its minimum value of - 2 at $x = - 1$
Find the difference between the maximum and minimum values of $( f ( x ) ) ^ { 2 } - f ( x )$

todai-math 2020 Q3 Determine intervals of increase/decrease or monotonicity conditions View

3

For real numbers $t$ satisfying $-1 \leq t \leq 1$, let $$x(t) = (1+t)\sqrt{1+t}, \quad y(t) = 3(1+t)\sqrt{1-t}$$ Consider the point $\mathrm{P}(x(t),\ y(t))$ in the coordinate plane.

(1) Show that the function $\dfrac{y(t)}{x(t)}$ of $t$ on $-1 < t \leq 1$ is strictly decreasing.

(2) Let $f(t)$ be the distance from the origin to $\mathrm{P}$. Investigate the monotonicity of the function $f(t)$ of $t$ on $-1 \leq t \leq 1$, and find its maximum value.

(3) Let $C$ be the locus of $\mathrm{P}$ as $t$ ranges over $-1 \leq t \leq 1$, and let $D$ be the region enclosed by $C$ and the $x$-axis. When $D$ is rotated $90°$ clockwise about the origin, find the area of the region swept out by $D$.
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todai-math 2021 Q5 Find absolute extrema on a closed interval or domain View

5

Let $\alpha$ be a positive real number. Define the function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ as the square of the distance AP between the two points $\mathrm{A}(-\alpha,\ -3)$ and $\mathrm{P}(\theta + \sin\theta,\ \cos\theta)$ in the coordinate plane.

[(1)] Show that there exists exactly one $\theta$ in the range $0 < \theta < \pi$ such that $f'(\theta) = 0$.
[(2)] Find the range of $\alpha$ such that the following holds: [6pt] The function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ attains its maximum at some point in the interval $0 < \theta < \dfrac{\pi}{2}$.

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todai-math 2022 Q1 Find absolute extrema on a closed interval or domain View

1

Consider the following function $f(x)$.
$$f(x) = (\cos x)\log(\cos x) - \cos x + \int_0^x (\cos t)\log(\cos t)\, dt \quad \left(0 \leq x < \frac{\pi}{2}\right)$$

[(1)] Show that $f(x)$ has a minimum value on the interval $0 \leq x < \dfrac{\pi}{2}$.
[(2)] Find the minimum value of $f(x)$ on the interval $0 \leq x < \dfrac{\pi}{2}$.

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todai-math 2024 Q2 Find absolute extrema on a closed interval or domain View

2

Consider the following function $f(x)$. $\displaystyle f(x) = \int_0^1 \frac{|t-x|}{1+t^2}\,dt \quad (0 \leq x \leq 1)$

(1) Find all real numbers $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan\alpha) = 0$.

(2) For the value of $\alpha$ found in (1), find the value of $\tan\alpha$.

(3) Find the maximum value and the minimum value of $f(x)$ on the interval $0 \leq x \leq 1$. If necessary, you may use the fact that $0.69 < \log 2 < 0.7$.
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turkey-yks 2010 Q41 Find absolute extrema on a closed interval or domain View

$$f(x) = x^{4} - 5x^{2} + 4$$
What is the maximum value of the function on the interval $\left[\frac{-1}{2}, \frac{1}{2}\right]$?
A) 8
B) 6
C) 4
D) 2
E) 0

turkey-yks 2011 Q31 Optimization Subject to an Algebraic Constraint View

For points $(x, y)$ on the boundary of the bounded region between the parabola $y = x ^ { 2 }$ and the line $y = 2 - x$, what is the maximum value of the expression $\mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } }$?
A) 25
B) 20
C) 17
D) 13
E) 10

turkey-yks 2011 Q43 MCQ on derivative and graph interpretation View

Below is the graph of the derivative of a function f defined on the interval $[ - 5,5 ]$.
According to this graph, I. The function f is decreasing for $x > 0$. II. $f ( - 2 ) > f ( 0 ) > f ( 2 )$. III. The function f has local extrema at $x = - 2$ and $x = 2$. Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) I and III
E) I, II and III

turkey-yks 2012 Q44 Determine parameters from given extremum conditions View

A third-degree real-coefficient polynomial function $P(x)$ with leading coefficient 1 has two of its roots as $-5$ and $2$.
If $P(x)$ has a local extremum at the point $x = 0$, what is the third root?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 7 } { 3 }$
D) $\frac { -5 } { 2 }$
E) $\frac { -10 } { 3 }$

turkey-yks 2012 Q45 MCQ on derivative and graph interpretation View

Below, the graph of the derivative of a function f that is defined and continuous on the set of real numbers is given.
Accordingly,
I. $f ( 2 ) - f ( 1 ) = -2$. II. The function f has a local maximum at the point $x = 0$. III. The second derivative function is defined at the point $x = 0$.
Which of the following statements are true?
A) Only I
B) Only III
C) I and II
D) II and III
E) I, II and III

turkey-yks 2013 Q44 Geometric or applied optimisation problem View

A tour company charges 140 TL per person for a tour it will organize. If the number of registered participants exceeds 80, a refund of 50 kuruş will be made to all participants for each person above 80. The capacity is limited to 200 people.
For example, if 100 people participate in the tour, everyone receives a 10 TL refund and the per-person fee is 130 TL.
Accordingly, how many people should participate in the tour so that the company's revenue from participants is maximum?
A) 160
B) 165
C) 175
D) 180
E) 185

turkey-yks 2015 Q42 Analyze function behavior from graph or table of derivative View

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given below.
Accordingly, regarding the function f: I. It is decreasing. II. $f ( a )$ is a local maximum value. III. $f ^ { \prime \prime } ( a )$ is not defined.
Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

turkey-yks 2016 Q44 Analyze function behavior from graph or table of derivative View

Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure.
Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value.
Which of the following statements are definitely true?
A) Only II
B) Only III
C) I and II
D) II and III
E) I, II and III

turkey-yks 2018 Q19 Analyze function behavior from graph or table of derivative View

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given in the following Cartesian coordinate plane.
Accordingly; what is the correct ordering of the values $\mathbf { f } ( \mathbf { 0 } )$, $\mathbf { f } ( \mathbf { 1 } )$ and $\mathbf { f } ( \mathbf { 2 } )$?
A) $\mathrm { f } ( 0 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 2 )$ B) $\mathrm { f } ( 0 ) < \mathrm { f } ( 2 ) < \mathrm { f } ( 1 )$ C) $f ( 1 ) < f ( 2 ) < f ( 0 )$ D) $\mathrm { f } ( 2 ) < \mathrm { f } ( 0 ) < \mathrm { f } ( 1 )$ E) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$

turkey-yks 2019 Q24 Chain Rule with Composition of Explicit Functions View

A function f is defined on the set of real numbers as
$$f ( x ) = x ^ { 2 } + x - 4$$
A function g defined and continuous on the set of real numbers has a derivative $g ^ { \prime }$ such that $g ^ { \prime } ( x ) = 0$ only for $x = 2$. Accordingly, the product of the x values satisfying
$$( g \circ f ) ^ { \prime } ( x ) = 0$$
is what?
A) 0
B) 1
C) 3
D) 4
E) 6

turkey-yks 2019 Q30 MCQ on derivative and graph interpretation View

A function f is continuous on the closed interval $[ 0,6 ]$ and differentiable on each of the open intervals $( 0,3 ) , ( 3,4 ) , ( 4,6 )$. The graph of its derivative $f ^ { \prime }$ is given in the rectangular coordinate plane below.
$$\begin{gathered} \text{Let } 0 < c < 2 \text{ and } \\ f ( 0 ) = 5 \end{gathered}$$
Accordingly, which of the following could be the value of f(6)?
A) 5,5
B) 7,3
C) 10,1
D) 12,7
E) 14,9

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49