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WINTER 2012 FINAL EXAM
Calculus for Electronics Engineering Technology
Dawson College: Department of Mathematics Date: May 22nd 2012, 9:30am to 12:30pm Course Code: 201-NYA-05 Section 6 Examiner: Emilie Richer
INSTRUCTIONS:
- All questions are to be answered directly on the examination paper in the space provided. If you need more space for your answer use the back of the page.
- SHOW ALL YOUR WORK: Show all your work clearly and justify all your answers.
- Verify that your final examination copy has a total of 19 pages including the cover page.
Question # Marks 1 10 2 6 3 5 4 5 5 5 6 5 7 5 8 5 9 5 10 12 11 10 12 12 13 5 14 5 15 5 Total 100
Question 1. (10 marks (1 mark each)) Differentiate the following functions with respect to x.
(a) f ( x ) = 4 cos( x^2 ) (f) f ( x ) = e sin^ x
(b) f ( x ) = 3 x^ + 2 x (g) f ( x ) = ln(2 sin x )
(c) f ( x ) = ( 2 x โ 1 )โ^2 (h) f ( x ) = tan( x + 2 )
(d) h ( x ) = 3 cos x โ^ x 1 (i) f ( x ) = (^) x^43
(e) f ( x ) = tan x^ x (j) f ( x ) = (^1) x โ 3 x โ^3
(d) โซ ex^ โ
x
dx
(e) โซ 2 x^3
(f)
โซ (^41) x (^3) โ 3 x (^2) + 1
x
dx
Question 3. (5 marks)
Sketch a graph that satisfies all of the following conditions:
x limโโ f^ ( x )^ โ^ โ
lim x โ 1 +^
f ( x ) โ โ
lim x โ 1 โ^
f ( x ) โ โโ
lim x โโ 1
f ( x ) = โ 1
f ( 2 ) = 0
lim x โ 2
f ( x ) does not exist
f ( 0 ) = 3
Question 5. (5 marks) Use the graph of y = f ( x ) pictured above to find the following values. If the value does not exist, write DNE.
-7 -6 -2 -1 (^1 2 3 4 5 6 )
1
2
3
4
5
6
7
x
y
-5 -4 -
(a) f ( 0 ) = _______ (h) lim x โโโ f ( x ) = _______
(b) lim x โ 5 +^
f ( x ) = _______ (i) f ( 5 ) = _______
(c) lim x โโ 5 +^
f ( x ) = _______ (j) lim x โ 0 โ^
f ( x ) = _______
(d) lim x โ 4
f ( x ) = _______ (k) f โฒ(โ 1 ) = _______
(e) lim x โ+โ f ( x ) = _______ (l)
โซ (^0)
โ 3
f ( x ) dx = _______
(f) lim x โ 5
f ( x ) = _______ (m)
โซ (^7)
5
f ( x ) dx = _______
(g) lim x โโ 3 โ^
f ( x ) = _______
Question 6. (5 marks)
Sketch the curves y = 2 cos x , y = 1 and find the area between them for 0 โค x โค ฯ.
Question 9. (5 marks)
Find the equation of the tangent line to the curve f ( x ) = e^2 x^ โ 3 x at the point ( 0 , 1 ).
Question 10. (12 marks (3 marks each)) Find the derivatives of the following functions.
(a) h ( t ) = e cos(^4 t )
(b) g ( z ) = 3 z โ^2 ln(sin z )
(c) f ( x ) = log 3 (tan( x^3 ))
(d) g ( x ) = ( 2 x โ 1 )(sin( 4 x ))( e โ x )
(d) The intervals where f ( x ) is concave up/down and any points of inflection
SKETCH OF f ( x ) = x^3 โ 3 x
Question 12. 12 marks (3 marks each) Integrate the following. (a) โซ (^) โ2 sin( 2 x )
cos 2 x
dx
(b) โซ ( 20 x^4 โ 18 x^2 )( 2 x^5 โ 3 x^3 )โ^8 dx
Question 13. (5 marks) A discharged ( Vc = 0 at t = 0) 4mF capacitor is to be charged by a current of i = 25 e^1 โ^0.^75 t^ mA. Find the capacitor voltage ( Vc ) at t = 135ms.
Question 14. (5 marks)
In the electric circuit shown below, the voltage E = 5 (in volts) and resistance r = 100 (in ohms) are constant, R is the resistance of a load.
E
_
r
R
In such a circuit, the electric current i is given by (^) r + ER and the power P delivered to the load R is given by P = Ri^2.
Given that R is positive, determine the value of R so that the power P delivered to R is a maximum.
BLANK PAGE
(Note: If you detach this page, make sure to hand it in with your exam)
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