16. If $f ( x ) = \sin \left( e ^ { - x } \right)$, then $f ^ { \prime } ( x ) =$ (A) $- \cos \left( e ^ { - x } \right)$ (B) $\cos \left( e ^ { - x } \right) + e ^ { - x }$ (C) $\cos \left( e ^ { - x } \right) - e ^ { - x }$ (D) $e ^ { - x } \cos \left( e ^ { - x } \right)$ (E) $- e ^ { - x } \cos \left( e ^ { - x } \right)$ [Figure]
Let $f$ be the real-valued function defined by $f ( x ) = \sqrt { 1 + 6 x }$. (a) Give the domain and range of $f$. (b) Determine the slope of the line tangent to the graph of $f$ at $x = 4$. (c) Determine the y -intercept of the line tangent to the graph of f at $\mathrm { x } = 4$. (d) Give the coordinates of the point on the graph of $f$ where the tangent line is parallel to $y = x + 12$.
The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $$f(0) = 2,\quad f'(0) = -4,\quad \text{and}\quad f''(0) = 3.$$ (a) The function $g$ is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $a$. Show the work that leads to your answers. (b) The function $h$ is given by $h(x) = \cos(kx)f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.
Let $f$ be the function given by $f ( x ) = \sqrt { x ^ { 4 } - 16 x ^ { 2 } }$. (a) Find the domain of $f$. (b) Describe the symmetry, if any, of the graph of $f$. (c) Find $f ^ { \prime } ( x )$. (d) Find the slope of the line normal to the graph of $f$ at $x = 5$.
Functions $f$, $g$, and $h$ are twice-differentiable functions with $g(2) = h(2) = 4$. The line $y = 4 + \dfrac{2}{3}(x - 2)$ is tangent to both the graph of $g$ at $x = 2$ and the graph of $h$ at $x = 2$. (a) Find $h'(2)$. (b) Let $a$ be the function given by $a(x) = 3x^3 h(x)$. Write an expression for $a'(x)$. Find $a'(2)$. (c) The function $h$ satisfies $h(x) = \dfrac{x^2 - 4}{1 - (f(x))^3}$ for $x \neq 2$. It is known that $\lim_{x \to 2} h(x)$ can be evaluated using L'H\^{o}pital's Rule. Use $\lim_{x \to 2} h(x)$ to find $f(2)$ and $f'(2)$. Show the work that leads to your answers. (d) It is known that $g(x) \leq h(x)$ for $1 < x < 3$. Let $k$ be a function satisfying $g(x) \leq k(x) \leq h(x)$ for $1 < x < 3$. Is $k$ continuous at $x = 2$? Justify your answer.
6. The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $$f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 4 , \text { and } f ^ { \prime \prime } ( 0 ) = 3$$ (a) The function $g$ is given by $g ( x ) = e ^ { a x } + f ( x )$ for all real numbers, where $a$ is a constant. Find $g ^ { \prime } ( 0 )$ and $g ^ { \prime \prime } ( 0 )$ in terms of $a$. Show the work that leads to your answers. (b) The function $h$ is given by $h ( x ) = \cos ( k x ) f ( x )$ for all real numbers, where $k$ is a constant. Find $h ^ { \prime } ( x )$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.
Let $f$ be a twice-differentiable function defined on the interval $- 1.2 < x < 3.2$ with $f ( 1 ) = 2$. The graph of $f ^ { \prime }$, the derivative of $f$, is shown above. The graph of $f ^ { \prime }$ crosses the $x$-axis at $x = - 1$ and $x = 3$ and has a horizontal tangent at $x = 2$. Let $g$ be the function given by $g ( x ) = e ^ { f ( x ) }$. (a) Write an equation for the line tangent to the graph of $g$ at $x = 1$. (b) For $- 1.2 < x < 3.2$, find all values of $x$ at which $g$ has a local maximum. Justify your answer. (c) The second derivative of $g$ is $g ^ { \prime \prime } ( x ) = e ^ { f ( x ) } \left[ \left( f ^ { \prime } ( x ) \right) ^ { 2 } + f ^ { \prime \prime } ( x ) \right]$. Is $g ^ { \prime \prime } ( - 1 )$ positive, negative, or zero? Justify your answer. (d) Find the average rate of change of $g ^ { \prime }$, the derivative of $g$, over the interval $[ 1,3 ]$.
The function $h : x \mapsto x \cdot \ln \left( x ^ { 2 } \right)$ is given with maximum domain $D _ { h }$. (1a) [2 marks] State $D _ { h }$ and show that for the term of the derivative function $h ^ { \prime }$ of $h$ the following holds: $h ^ { \prime } ( x ) = \ln \left( x ^ { 2 } \right) + 2$. (1b) [3 marks] Determine the coordinates of the local maximum point of the graph of $h$ located in the second quadrant. Figure 1 shows the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ of a polynomial function $f$ defined on $\mathbb { R }$. Only at the points $\left( - 4 \mid f ^ { \prime } ( - 4 ) \right)$ and $\left( 5 \mid f ^ { \prime } ( 5 ) \right)$ does the graph $G _ { f ^ { \prime } }$ have horizontal tangents. [Figure] (2a) [2 marks] Justify that $f$ has exactly one inflection point. (2b) [2 marks] There are tangents to the graph of $f$ that are parallel to the angle bisector of the first and third quadrants. Using the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ in Figure 1, determine approximate values for the $x$-coordinates of those points where the graph of $f$ has such a tangent. The functions $f : x \mapsto x ^ { 2 } + 4$ and $g _ { m } : x \mapsto m \cdot x$ with $m \in \mathbb { R }$ are given, both defined on $\mathbb { R }$. The graph of $f$ is denoted by $G _ { f }$ and the graph of $g _ { m }$ by $G _ { m }$. (3a) [3 marks] Sketch $G _ { f }$ in a coordinate system. Calculate the coordinates of the common point of the graphs $G _ { f }$ and $G _ { 4 }$. (3b) [2 marks] There are values of $m$ for which the graphs $G _ { f }$ and $G _ { m }$ have no common point. State these values of $m$. The function $g$ is given by $g ( x ) = 0,7 \cdot e ^ { 0,5 x } - 0,7$ with $x \in \mathbb { R }$. The function $g$ is invertible. Figure 2 shows the graph $G _ { g }$ of $g$ and part of the graph $G _ { h }$ of the inverse function $h$ of $g$. [Figure] (4a) [2 marks] Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$. Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$. The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]
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 $\varphi'(t)$.
118- If $f(x) = \dfrac{x^3 - 2}{1 + x^3}$, $g(x) = \sqrt[3]{x-1}$, then $f'(g(x)) \cdot g'(x)$ is equal to which of the following? (1) $\dfrac{3}{x}$ (2) $\dfrac{3}{x^2}$ (3) $\dfrac{1}{3x}$ (4) $\dfrac{x-3}{x^2}$
112. The derivative of the function $y = \cos^2(\tan^{-1}x)$, at $x = 1$, is which of the following? (1) $-\dfrac{1}{2}$ (2) $-\dfrac{1}{4}$ (3) $\dfrac{1}{4}$ (4) $1$
112- The derivative of $f(x) = \sin\!\left(\dfrac{\pi}{2} + \tan^{-1}\dfrac{x}{2}\right)$ at the point $x = 2\sqrt{3}$ is which of the following? (1) $-\dfrac{1}{24}$ (2) $-\dfrac{1}{16}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{1}{4}$
If $f^{\prime}(x) = \sin(\log x)$ and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to (the question continues with answer options as given in the paper).
Let $f$ be a differentiable function such that $f ( 1 ) = 2$ and $f ^ { \prime } ( x ) = f ( x )$ for all $x \in R$. If $h ( x ) = f ( f ( x ) )$, then $h ^ { \prime } ( 1 )$ is equal to : (1) $4 e ^ { 2 }$ (2) $2 e$ (3) $4 e$ (4) $2 e ^ { 2 }$
If $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = 3$, then the derivative of $f ( f ( f ( x ) ) ) + ( f ( x ) ) ^ { 2 }$ at $x = 1$ is: (1) 9 (2) 12 (3) 15 (4) 33