b) $\lim_{x \to 1} \frac{\sin(x) - \sin(1)}{x^2 - 1}$

c) $\lim_{x \to \infty} \frac{7x - 8x^2 + 5x^3}{2x^3 - x^2 + 6x - 1}$

2. (15 points) Consider the function $f(x) = e^{x - x^2}$.

a) Find the domain and the equations of vertical and horizontal asymptotes.

a) Find the domain and the equations of vertical and horizontal asymptotes.

b) Find the first derivative and determine the intervals where the function is increasing/decreasing.

c) Find the second derivative and determine the intervals where the function is concave up/down.

d) Sketch the graph of this function.

3. (15 points) Compute the derivatives of the following functions:

a) $f(x) = \sin(x^3 - x)$

a) $f(x) = \sin(x^3 - x)$

c) $f(x) = \frac{\sin x}{x + 1}$

4. (15 points) Find the equation of the tangent line to the graph of the function $f(x) = \ln(2x + 3)$ at $x = -1$.

5. (15 points) Find $y'$ given the equation $x^3 + y^3 + x + y = 1$.

6. (15 points) Find the minimal and maximal values of the function
$f(x) = x^4 + 3x^2 + 5$
on the interval $[-2, 2]$.

7. (10 points) Find the maximal area of a rectangle that fits between the graph of the function
$f(x) = \frac{1}{x^2 + 1}$
and the $x$-axis. Assume that one side of the rectangle belongs to the $x$-axis.

8. (10 points) (Bonus Problem) Find the number of solutions to the equation
$x^3 + 2x^2 - x - 1 = 0$
Justify your answer.

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