$$f ( x ) = \begin{cases} \frac { 2 } { x } & \text { for } x < - 1 \\ x ^ { 2 } - 3 & \text { for } - 1 \leq x \leq 2 \\ 4 x - 3 & \text { for } x > 2 \end{cases}$$ Let $f$ be the function defined above. At what values of $x$, if any, is $f$ not differentiable? (A) $x = - 1$ only (B) $x = 2$ only (C) $x = - 1$ and $x = - 2$ (D) $f$ is differentiable for all values of $x$.
18. An equation of the line tangent to the graph of $y = x + \cos x$ at the point $( 0,1 )$ is (A) $y = 2 x + 1$ (B) $y = x + 1$ (C) $y = x$ (D) $y = x - 1$ (E) $y = 0$
Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$. (a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$. (b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer. (c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$. (d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.
The graph of $y = e ^ { \tan x } - 2$ crosses the $x$-axis at one point in the interval $[ 0,1 ]$. What is the slope of the graph at this point? (A) 0.606 (B) 2 (C) 2.242 (D) 2.961 (E) 3.747
Let $f$ be the function defined by $f ( x ) = e ^ { x } \cos x$. (a) Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$. (b) What is the slope of the line tangent to the graph of $f$ at $x = \frac { 3 \pi } { 2 }$ ? (c) Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$. Justify your answer. (d) Let $g$ be a differentiable function such that $g \left( \frac { \pi } { 2 } \right) = 0$. The graph of $g ^ { \prime }$, the derivative of $g$, is shown below. Find the value of $\lim _ { x \rightarrow \pi / 2 } \frac { f ( x ) } { g ( x ) }$ or state that it does not exist. Justify your answer.
Let $f$ be the function defined by $f(x) = \frac{3}{2x^2 - 7x + 5}$. (a) Find the slope of the line tangent to the graph of $f$ at $x = 3$. (b) Find the $x$-coordinate of each critical point of $f$ in the interval $1 < x < 2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers. (c) Using the identity that $\frac{3}{2x^2 - 7x + 5} = \frac{2}{2x - 5} - \frac{1}{x - 1}$, evaluate $\int_{5}^{\infty} f(x)\, dx$ or show that the integral diverges. (d) Determine whether the series $\sum_{n=5}^{\infty} \frac{3}{2n^2 - 7n + 5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.
Question 1: Consider the function $g$ defined on $]0;+\infty[$ by $g(x) = x^2 + 2x - \frac{3}{x}$. An equation of the tangent line to the curve representing $g$ at the point with abscissa 1 is:
The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$. We can assert that: a. $f ^ { \prime } ( - 0.5 ) = 0$ b. if $x \in ] - \infty ; - 0.5 \left[ \right.$, then $f ^ { \prime } ( x ) < 0$ c. $f ^ { \prime } ( 0 ) = 15$ d. the derivative function $f ^ { \prime }$ does not change sign on $\mathbb { R }$.
3. We consider the function $f$ defined on $] 0 ; + \infty \left[ \right.$ whose representative curve $C _ { f }$ is given in an orthonormal coordinate system in the figure (Fig. 1) on page 5. We specify that:
$T$ is the tangent to $C _ { f }$ at point $A$ with abscissa 8;
The $x$-axis is the horizontal tangent to $C _ { f }$ at the point with abscissa 1.
[Figure] Fig. 1 Statement 3: According to the graph, the function $f$ is convex on its domain of definition.
The equation of the tangent line to the curve $y = - x ^ { 3 } + 4 x$ at the point $( 1,3 )$ is $y = a x + b$. Find the value of $10 a + b$. (where $a , b$ are constants) [4 points]
Two polynomial functions $f ( x ) , g ( x )$ satisfy the following conditions. (가) $g ( x ) = x ^ { 3 } f ( x ) - 7$ (나) $\lim _ { x \rightarrow 2 } \frac { f ( x ) - g ( x ) } { x - 2 } = 2$ When the equation of the tangent line to the curve $y = g ( x )$ at the point $( 2 , g ( 2 ) )$ is $y = a x + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (where $a , b$ are constants.) [4 points]
18. (This question is worth 13 points) Given the function $f ( x ) = \ln \frac { 1 + x } { 1 - x }$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0 , f ( 0 ) )$; (II) Prove: When $x \in ( 0,1 )$, $f ( x ) > 2 \left( x + \frac { x ^ { 3 } } { 3 } \right)$; (III) Let the real number $k$ be such that $f ( x ) > k \left( x + \frac { x ^ { 3 } } { 3 } \right)$ holds for all $x \in ( 0,1 )$. Find the maximum value of $k$.
(12 points) Let $A$ and $B$ be two points on the curve $C: y = \frac{x^2}{4}$, and the sum of the $x$-coordinates of $A$ and $B$ is 4. (1) Find the slope of line $AB$; (2) Find the equation of line $AB$.
10. The equation of the tangent line to the curve $y = 2 \sin x + \cos x$ at the point $( \pi , - 1 )$ is A. $x - y - \pi - 1 = 0$ B. $2 x - y - 2 \pi - 1 = 0$ C. $2 x + y - 2 \pi + 1 = 0$ D. $x + y - \pi + 1 = 0$
The equation of the tangent line to the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 }$ at the point $( 1 , f ( 1 ) )$ is A. $y = - 2 x - 1$ B. $y = - 2 x + 1$ C. $y = 2 x - 3$ D. $y = 2 x + 1$
119- If $f(x) = xe^x$; $x > 0$, then the tangent line to the graph of $f^{-1}$ at points located along $e$, intersects the $y$-axis at which value? (1) $\dfrac{1}{4}$ (2) $\dfrac{1}{3}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{1}{e}$
117. If $\theta$ is the angle between the left and right tangents to the graph of the function $f(x) = \left[x + \frac{1}{2}\right]x + x^2$, at the point $x = \frac{1}{2}$, what is $\tan\theta$? (1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$
119- The line $y = 3x - 2$ at the point $x = 2$ is tangent to the curve $y = f(x)$. What is $\displaystyle\lim_{x \to 2} \dfrac{f^2(x) - 4f(x)}{x - 2}$? (1) $2$ (2) $6$ (3) $12$ (4) $15$
117. If $\theta$ is the angle between the left and right tangents to the graph of $y = |\ln x|$ at the corner point, then $\tan\theta$ equals: (1) $-1$ (2) $1$ (3) zero (4) $\infty$
119. The function $f(x) = x + \ln x$ is defined (given). The equation of the tangent line to the graph of $f^{-1}$ at the point where it meets the bisector of the first quadrant is: (1) $y + 2x = 3$ (2) $2x - y = 1$ (3) $2x + y = 3$ (4) $2y - x = 1$
120. The $x$-intercept of the normal line to the curve $x^2 + y^2 = 3xy + 3$ at the point $(1, 2)$ is: (1) $2$ (2) $3$ (3) $4$ (4) $5$
117- A line is tangent to the graph of the function $y = x^3 - 2x^2 + 3x$ at the point $x = 2$ and passes through it. The slope of this line is which of the following? (1) $-\dfrac{2}{3}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{5}{3}$ %% Page 5 Mathematics120-CPage 4
Determine the equation of the tangent line to the curve with equation $y = \sqrt{25 - x^{2}}$ at its point with abscissa 3, using two different methods.