https://file.aiquickdraw.com/mathbot/file/1729951390221lim9sg6l.pdf

gör andra uppgifter i samma struktur av denna exempeltenta

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.

Explore sample problems for Mathematical Analysis 1, covering algebra, limit computations, differentiation, and analytic geometry. This example-based guide provides tasks that test core concepts such as solving equations, calculating limits, finding tangent lines, and analyzing function behavior.

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