LFM Pure

ap-calculus-ab None Q1 Limit Involving Derivative Definition of Composed Functions View

$\lim _ { x \rightarrow \pi } \frac { \cos x + \sin ( 2 x ) + 1 } { x ^ { 2 } - \pi ^ { 2 } }$ is
(A) $\frac { 1 } { 2 \pi }$
(B) $\frac { 1 } { \pi }$
(C) 1
(D) nonexistent

ap-calculus-ab None Q7 Chain Rule Combined with Fundamental Theorem of Calculus View

If $f ( x ) = \int _ { 1 } ^ { x ^ { 3 } } \frac { 1 } { 1 + \ln t } d t$ for $x \geq 1$, then $f ^ { \prime } ( 2 ) =$
(A) $\frac { 1 } { 1 + \ln 2 }$
(B) $\frac { 12 } { 1 + \ln 2 }$
(C) $\frac { 1 } { 1 + \ln 8 }$
(D) $\frac { 12 } { 1 + \ln 8 }$

ap-calculus-ab 2007 Q3 Chain Rule with Table-Defined Functions View

The functions $f$ and $g$ are differentiable for all real numbers, and $g$ is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of $x$.

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
1	6	4	2	5
2	9	2	3	1
3	10	-4	4	2
4	-1	3	6	7

The function $h$ is given by $h(x) = f(g(x)) - 6$.
(a) Explain why there must be a value $r$ for $1 < r < 3$ such that $h(r) = -5$.
(b) Explain why there must be a value $c$ for $1 < c < 3$ such that $h^{\prime}(c) = -5$.
(c) Let $w$ be the function given by $w(x) = \int_{1}^{g(x)} f(t)\, dt$. Find the value of $w^{\prime}(3)$.
(d) If $g^{-1}$ is the inverse function of $g$, write an equation for the line tangent to the graph of $y = g^{-1}(x)$ at $x = 2$.

ap-calculus-ab 2012 Q1 Straightforward Polynomial or Basic Differentiation View

If $y = x \sin x$, then $\frac { d y } { d x } =$
(A) $\sin x + \cos x$
(B) $\sin x + x \cos x$
(C) $\sin x - x \cos x$
(D) $x ( \sin x + \cos x )$
(E) $x ( \sin x - \cos x )$

ap-calculus-ab 2012 Q7 Chain Rule with Composition of Explicit Functions View

If $y = \left( x ^ { 3 } - \cos x \right) ^ { 5 }$, then $y ^ { \prime } =$
(A) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 }$
(B) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 }$
(C) $5 \left( 3 x ^ { 2 } + \sin x \right)$
(D) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 } \cdot ( 6 x + \cos x )$
(E) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 } \cdot \left( 3 x ^ { 2 } + \sin x \right)$

ap-calculus-ab 2012 Q14 Chain Rule with Composition of Explicit Functions View

If $f ( x ) = \sqrt { x ^ { 2 } - 4 }$ and $g ( x ) = 3 x - 2$, then the derivative of $f ( g ( x ) )$ at $x = 3$ is
(A) $\frac { 7 } { \sqrt { 5 } }$
(B) $\frac { 14 } { \sqrt { 5 } }$
(C) $\frac { 18 } { \sqrt { 5 } }$
(D) $\frac { 15 } { \sqrt { 21 } }$
(E) $\frac { 30 } { \sqrt { 21 } }$

ap-calculus-bc 2012 Q1 Chain Rule with Composition of Explicit Functions View

If $y = \sin ^ { 3 } x$, then $\frac { d y } { d x } =$
(A) $\cos ^ { 3 } x$
(B) $3 \cos ^ { 2 } x$
(C) $3 \sin ^ { 2 } x$
(D) $- 3 \sin ^ { 2 } x \cos x$
(E) $3 \sin ^ { 2 } x \cos x$

ap-calculus-bc 2012 Q77 Straightforward Polynomial or Basic Differentiation View

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = x ^ { 4 }$. On what intervals is the rate of change of $f ( x )$ greater than the rate of change of $g ( x )$ ?
(A) $( 0.831, 7.384 )$ only
(B) $( - \infty , 0.831 )$ and $( 7.384 , \infty )$
(C) $( - \infty , - 0.816 )$ and $( 1.430, 8.613 )$
(D) $( - 0.816, 1.430 )$ and $( 8.613 , \infty )$
(E) $( - \infty , \infty )$

cmi-entrance 2022 QA7 4 marks Iterated/Nested Exponential Differentiation View

Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$. (26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$. (27) $\int_0^1 f_3(x)\, dx = 1$. (28) The derivative of $f_{123}$ at $x = 1$ is 1.

csat-suneung 2005 Q18 3 marks Limit Involving Derivative Definition of Composed Functions View

Two real numbers $a$ and $b$ satisfy $\lim _ { x \rightarrow 2 } \frac { \sqrt { x ^ { 2 } + a } - b } { x - 2 } = \frac { 2 } { 5 }$. Find the value of $a + b$. [3 points]

csat-suneung 2005 Q28 4 marks Geometric Limit with Parametric Chain Rule View

For two points $\mathrm { P } ( n , f ( n ) )$ and $\mathrm { Q } ( n + 1 , f ( n + 1 ) )$ on the graph of the quadratic function $f ( x ) = 3 x ^ { 2 }$, let $a _ { n }$ be the distance between them. Find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$. (Here, $n$ is a natural number.) [4 points]
(1) 9
(2) 8
(3) 7
(4) 6
(5) 5

csat-suneung 2006 Q18 3 marks Limit Involving Derivative Definition of Composed Functions View

For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]

csat-suneung 2009 Q29 4 marks Derivative of Inverse Functions View

(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$

csat-suneung 2010 Q3 2 marks Continuity Conditions via Composition View

For two constants $a , b$, if $\lim _ { x \rightarrow 3 } \frac { \sqrt { x + a } - b } { x - 3 } = \frac { 1 } { 4 }$, what is the value of $a + b$? [2 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

csat-suneung 2010 Q27 3 marks Geometric Limit with Parametric Chain Rule View

[Calculus] As shown in the figure, let Q be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point P on the circle meets the $x$-axis. For point $\mathrm { A } ( - 1,0 )$ and the origin O, let $\angle \mathrm { PAO } = \theta$. Find the value of $\lim _ { \theta \rightarrow \frac { \pi } { 4 } - 0 } \frac { \overline { \mathrm { PQ } } - \overline { \mathrm { OQ } } } { \theta - \frac { \pi } { 4 } }$. (where point P is in the first quadrant) [3 points]
(1) 2
(2) $\sqrt { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 1
(5) $\frac { \sqrt { 2 } } { 2 }$

csat-suneung 2012 Q22 3 marks Limit Evaluation Involving Composition or Substitution View

Find the value of $\lim _ { x \rightarrow 1 } \frac { ( x - 1 ) \left( x ^ { 2 } + 3 x + 7 \right) } { x - 1 }$. [3 points]

csat-suneung 2016 Q5 3 marks Straightforward Polynomial or Basic Differentiation View

For the function $f ( x ) = x ^ { 3 } + 7 x + 3$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

csat-suneung 2018 Q9 3 marks Limit Involving Derivative Definition of Composed Functions View

For a function $f ( x )$ differentiable on the set of all real numbers, let the function $g ( x )$ be defined as $$g ( x ) = \frac { f ( x ) } { e ^ { x - 2 } }$$ If $\lim _ { x \rightarrow 2 } \frac { f ( x ) - 3 } { x - 2 } = 5$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2018 Q23 3 marks Straightforward Polynomial or Basic Differentiation View

For the function $f ( x ) = 2 x ^ { 3 } + x + 1$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

csat-suneung 2022 Q3 2 marks Straightforward Polynomial or Basic Differentiation View

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 1$, what is the value of $f ^ { \prime } ( 1 )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

gaokao 2015 Q11 Straightforward Polynomial or Basic Differentiation View

11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.

gaokao 2019 Q13 Chain Rule with Composition of Explicit Functions View

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$.

gaokao 2019 Q13 Chain Rule with Composition of Explicit Functions View

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$ .

grandes-ecoles 2011 Q6 Higher-Order Derivatives of Products/Compositions View

Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$

grandes-ecoles 2014 QIB1 Chain Rule with Composition of Explicit Functions View

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144