The provided file is an example exam ("Exempeltenta") for a course in Mathematical Analysis 1. It consists of two parts, each containing various types of problems. I'll suggest some tasks that follow a similar structure as the questions in the example exam.

New Example Tasks

Del 1

1. a) Solve the equation $2(x + 3) = 3(4 - \frac{x}{2})$ (2p)

   b) Calculate $\frac{2^{10}}{2^{3}} \cdot (4^{-2})$ (2p)

2. Solve the equations:

   a) $\frac{x - 4}{x + 1} = 0$ (2p)

   b) $\sqrt{2x + 1} = x + 3$ (2p)

3. Let $f(x) = 2x - 5$, $g(x) = \ln(\sqrt{x})$.

   a) Determine the domain of $g(x)$. (1p)

   b) Find $g \circ f(x)$ and $f^{-1}(x)$. (2p)

4. a) Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$. (2p)

   b) Differentiate the function $f(x) = \sqrt{4x + 7}$. (2p)

5. The rational function $f(x) = \frac{x^2 - 4x + 3}{x - 2}$ is given.

   a) Find all stationary points and local extrema of $f$. (3p)

   b) Determine any horizontal, vertical, or oblique asymptotes of $y = f(x)$. (2p)

   c) Sketch the graph of the function, marking any local maxima, minima, inflection points, and asymptotes. (2p)

6. Find the equation of the tangent to the curve $y = x \sin x$ at the point where $x = \frac{\pi}{4}$. (3p)

Del 2

7. A circle is given by the equation $x^2 + y^2 - 4x - 6y = 0$ and a line by $y = -x + 2$.

   a) Determine the center and radius of the circle, then draw the circle and the line in a coordinate system. (2p)

   b) Calculate the exact points of intersection between the circle and the line. (3p)

8. a) For which $x$ is the function $f(x) = \ln(x^2 - 1) - \ln(x - 1)$ undefined? (2p)

   b) Solve the inequality $\ln(x^2 - 1) - \ln(x - 1) \leq 0$. (3p)

9. Determine the maximum and minimum values of the function

   $f(x) = 2x - 3\sin x$

   in the interval $[0, 2\pi]$. (5p)

Tip:

Make sure to practice differentiation and limit problems, as these are commonly tested in mathematical analysis exams.