EINDHOVEN University of Technology Department of Mathematics and Computer Science

Examination Basic Mathematics, 2DL03, Monday 10 November 2008, 9.00– 12.00. Write clearly the program (Pre-master program or HBO-minor) you are following on the first page of your work. If you don’t have an identity number, write on the first paper behind Ident.nr: NONE. The exam consists of two parts, a Common part and a Pre-master part/Part doorstroomminor HBO. The Common part has to be done in two hours and contains 9 problems, for which you can get 40 points. The Pre-master part/Part doorstroomminor HBO has to be done in one hour and contains 5 problems, for which you can get 20 points. For information about the partition of the points over the exercises, see at the end. If you are making both parts, you get two marks, one for the first part and one for the entire exam. All students have to make the Common part. HBO-students, which want to do the minor Academic orientation, only have to do the first part. These students have three hours to make the exam. All other students have also to do the Pre-master part/Part doorstroomminor HBO. For HAN-students: the Common part is level 3 and the whole examination is level 4. Formulate the computations and the results of the exercises in a clear way. It is not allowed to use a laptop, graphical calculator or chart with formulas. It is not allowed to use a book or other hand-written material.

Common part 2x x−3 + ≤ 2. x−3 2x √ 2. Find the Taylor polynomial of order three about x = 3 for f (x) = x + 1. Give the equation of the tangent line to the graph of f through the point (3,2). 1. Determine all x in R for which the following inequality holds

3. Find an equation of the tangent line to the ellipse given by 2x2 + 3y 2 = 5 at the point P(1,1). Also give the equation of the normal in P.

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Examination Basic Mathematics, 2DL03, Monday 10 November 2008, 9.00–12.00

4. Sketch the set of points (x, y) in R2 which satisfy the inequalities x2 + y 2 ≤ 4x + 6y ´and x ≥ y . 5. Let ϑ ∈ ( 12 π, π) and sin(ϑ) = 13 . Compute cos(ϑ) en tan(2ϑ). 6. Consider the function f with f (x) = a) Find all the asymptotes. b) Find the inverse function f −1 .

2x−5 . x+2

7. Find the limits: x2 − 9 . a) lim x→3 2x − 6 x3 − x2 − 7x + 7 b) lim . x→1 x−1 Z (2 + ln(x))3 8. Calculate dx . x Z 9. Calculate xe5x dx.

Pre-master part/Part doorstroomminor HBO Z 10. Calculate 0

1 2

arctan(2x) dx. 1 + 4x2

1 11. Show that ln(5 + x) − ln(5) < x for all x > 0. 5 12. Consider the function f with the Taylor polynomial of order 3 around 2 given by p3 (x) = 7 + 2(x − 2) − 3(x − 2)2 + 5(x − 2)3 . (a) Calculate f (2), f 0 (2), f 00 (2), f (3) (2) and approximate f (2.1). (b) If 0 < f (4) (x) ≤ 12 on the intervall [2 , 2.1], give an estimation of the error E3 (2.1). It is known that f (2.1) = p3 (2.1) + E3 (2.1).

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Examination Basic Mathematics, 2DL03, Monday 10 November 2008, 9.00–12.00 Z 13. Calculate

x2 cos x dx.

14. Show that cos(2 arcsin(x)) = 1 − 2x2 .

For the problems you can get the following points: Common Problem Problem Problem Problem

part 1: 5 2: 6 3: 4 4: 4

points points points points

Problem Problem Problem Problem

5: 4 points 6a: 2 points 6b: 3 points 7a: 2 points

Problem Problem Problem

7b: 2 points 8: 4 points 9: 4 points

The mark for the Common part is obtained as follows: the total of the scored points is divided by 4 and is rounded off to the closest natural number. Pre-master part/Part doorstroomminor HBO Problem 10: 4 points Problem 12a: 3 points Problem 11: 4 points Problem 12b: 2 points

Problem 13: 4 points Problem 14: 3 points

The mark for the whole exam is obtained as follows: the total of the scored points of both parts is divided by 6 and is rounded off to the closest natural number.

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Integral Table g(x) xn , n 6= −1

R

1 x0 f (x) f (x) x

ln(|x|) ln(|f (x)|) ex

e ax , a > 0, a 6= 1 sin(x) cos(x)

g(x)dx

xn+1 n+1

ax ln(a)

− cos(x) sin(x) 1 − cot(x) sin2 (x) 1 tan(x) cos2 (x) tan(x) − ln(| cos(x)|) 1 ln(| tan( x2 )|) sin(x) 1 ln(| tan( x2 + π4 )|) cos(x) ax eax sin(bx), a2 + b2 > 0 a2e+b2 (a sin(bx) − b cos(bx)) ax eax cos(bx), a2 + b2 > 0 a2e+b2 (a cos(bx) + b sin(bx)) 1 1 x ,a > 0 arctan( ) a2 +x2 a a 1 1 a+x ,a > 0 ln( a−x ) a2 −x2 2a √ 1 ,a > 0 arcsin( xa ) a2 −x2 √ √ 1 ,a > 0 ln(x + x2 + a2 ) a2 +x2 √ x + x 2 − a2 ) √ 1 ln( , a > 0 2 2 √ √x −a 2 x x 2 − x2 , a > 0 a a2 − x2 + a2 arcsin( √ ) 2 √ a √ x a2 2 2 2 2 2 2 a + x + 2 ln(x 2 √ + √x + a ) √a + x , a > 0 2 a x x 2 − a2 , a > 0 x2 − a2 + 2 ln( x + x2 − a2 ) 2 sinh(x) cosh(x) cosh(x) sinh(x) tanh(x) ln(cosh(x)) Remarks All parameters are real numbers. The constants of integration have been omitted.

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Taylor Polynomials Function Taylor polynomial plus O-term ex cos(x)

1 2 1 x + · · · + xn + O(xn+1 ) 2 n! 1 1 4 (−1)n 2 n 1 − x2 + x + ··· + x + O(x2 n+1 ) 2 24 (2 n)!

1+x+

1 3 1 5 (−1)n x2 n+1 x + x + ··· + + O(x2 n+2 ) 6 120 (2 n + 1)!

sin(x)

x−

1 1+x

1 − x + x2 + · · · + (−1)n xn + O(xn+1 )

ln(1 + x) x − 1 1 + x2

1 2 1 3 (−1)n n+1 x + x + ··· + x + O(xn+2 ) 2 3 n+1

1 − x2 + x4 + · · · + (−1)n x2 n + O(x2 n+1 ) 1 3 1 5 (−1)n 2 n+1 x + x + ··· + x + O(x2 n+2 ) 3 5 (2 n + 1)       α α α 2 1+ x+ x + ··· + xn + O(xn+1 ) 1 2 n

arctan(x) x − (1 + x)α

• All Taylor polynomials are polynomials around the point 0. • The binomial coefficients are defined by   α · (α − 1) · (α − 2) · · · (α − (k − 1)) α = , k = 1, 2, 3, . . . k 1 · 2 · 3 ··· k   α =1 0

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