Description

Exam
Jacobs UniversityTime: 75 minutes

Name: Matriculation ID:
INSTRUCTIONS
• Make sure to write your name and ID on the first page and every page thereafter.
• The question booklet consists of 6 pages. Make sure you have all of them.
• Regarding question 5, you should only do two out of three options. You can pick the ones you prefer. Clearly state which ones they are.
• Keep quiet during the exam. For assistance, raise your hand and an invigilator will come to see you
• Answer the questions in the spaces provided after each question. If you run out of room for an answer, continue on the back of the page.
• The mark of each question is printed next to it.
• Use of mobile phones or other unauthorized electronic devices or material in the exam room is prohibited. Only simple calculators are allowed during the exam.
Question 1 2 3 4 5 Total
Points 18 16 20 32 14 100
Score

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Good luck!
Calculus and Linear Algebra I, Part I
(18 points)
Compute the following limits, if they exist. Else, argue why the limit does not exist.
4
7
7
(16 points)
6 (a) Show that the equation x6 − 5x − 5 = 0 has at least one solution on the interval
[−1,0].
10 (b) Compute the derivative of directly from its definition as the limit of a
difference quotient.

(20 points)
Consider the function .
What is the domain of f? Find the horizontal and vertical asymptotes, local minima, local maxima, and reflection points of f. Identify the regions where the graph of f is concave up or concave down. Finally, sketch the graph.
(32 points)
Solve the following:
10 (a) Integrate
R −3e−1/x2dx
6 (b) Integrate x
8 (c) Integrate
8 (d) Differentiate f(t) = tt3
(14 points)
Choose only two of the below:
7 (a) Find the area between the curves x = y2 and 0 = −x − y2 + 2 (in absolute terms).
7 (b) A farmer owns an 8km long stretch of land between two parallel rivers that are 1500m apart. What is the area of the largest rectangular enclosure he can fence off with (i) 1km of fencing and (ii) 4km of fencing, assuming that no fence is needed along the rivers?
7 (c) Use implicit differentiation to find an equation for the tangent line to the graph of
sin(2x + y) = y3 sin(x) at the point (0,0).