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NAME:
STUDENT ID:
DAWSON COLLEGE - DEPARTMENT OF MATHEMATICS
FINAL EXAMINATION
Calculus I - 201-NYA-05 Section: 03-
Instructor: O.Veres May 14, 2012 (6:30 p.m. - 9:30p.m.)
(MARKS)
(16) 1. Evaluate the following limits.
(a) lim x→∞
2 x^4 − 3 x− 2 4 x^4 +x^2 − 5 x+
(b) lim x→ 1
√ x+3− 2 1 −x
(c) lim x→ 0
x^2 −cos x+ 2 sin x−x+1−ex
(d) lim x→ 2
3 x^2 − 6 x |x− 2 |
(6) 2. Given f (x) =
k^2 x^2 − kx if x ≤ 3
kx − 1 if x > 3
Find all values of k such that f is continuous at x = 3. Justify your answer using the definition of continuity.
(6) 3. (a) Using only the definition of derivative, find f ′(x) for f (x) = 3 x+
(b) Check your answer using the derivative rules.
(6) 4. If y = f (x) satisfies the equation y 3 − 2 y 2
(a) Using implicit differentiation find y′^ at the point P (2, 1).
(b) Find an equation of the tangent line to the graph of y = f (x) at P (2, 1).
(16) 5. Differentiate each function.
(a) f (x) =
1 − 9 x^2 arccos(3x)
(b) f (x) = esin(3x)^ csc(x^2 )
(c) g(x) = tan(sin(x^2 )) + tan(x^2 ) sin(x^2 )
(d) y =
x 2
)√x
(4) 6. Find f ′(0), if
f (x) = ln
e^3 x
x^2 − 1
cos x √ 3 x^3 + 1
(6) 7. A plane flying horizontally at an altitude of 2km and a speed of
325 km/hr passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 5km away from the station.
(4) 8. Find the absolute minimum and maximum values of f (x) = x^2 x^2 +1 on [1, 2]
(14) 9. Given f (x) = x
(^2) − 3 x^3 , f ′(x) = (9−x^2 ) x^4 and f ′′(x) = 2(x^2 −18) x^5
Find
(a) The domain of f and the x and the y intercepts
(b) Horizontal and vertical asymptotes
(c) The intervals where f (x) is increasing or decreasing.
(d) Local maximum and minimum values of f (x)
(e) Intervals where f (x) is concave up or concave down
(f) Inflection points
(g) Using the above information sketch the graph of f (x). Clearly label the important points on the graph.
(6) 10. A box with a square base and an open top must have a volume of
256 m^3. Find the dimensions of the box that minimize the amount of material used.
ANSWERS:
1 (a) 1 2 (b)^ −
1 4 (c)^ −^3 (d) the limit does not exit 2 k = 1 3 3 f ′(x) = −^3 (x+1)^2
4 (a) y′^ = − 3 y 3 y^2 − 4 y+3x and^ y
′ = −^3
5 when^ x^ = 2,^ y^ = 1^ (b)^ y^ =^ −
3 5 x^ +^
11 5
5 (a) f ′(x) = √−^9 x 1 − 9 x^2
(arccos 3x) − 3
(b) f ′ (x) = csc(x 2 )e sin 3x (3 cos(3x) − 2 x cot(x 2 ))
(c) g ′ (x) = 2x(cos(x 2 ) sec 2 (sin(x 2 ))) + sin(x 2 ) sec 2 (x 2 ) + cos(x 2 ) tan(x 2 )
(d) y ′ = (x 2
√ x
ln(x^2 +1) 2
√ x +^
2 x √ x x^2 +
6 f ′(0) = 3
7 297. 867 km/hr
8 f (1) = 12 is absolute maximum f (2) = 45 is absolute minimum
9 (a) Df = (∞, 0) ∪ (0, ∞)
(b) x = 0 is vertical asymptote of y = f (x) to ±∞ y = 0 is horizontal asymptote of y = f (x) to ±∞
(c) f is decreasing on (−∞, −3)∪(3, ∞) and f is increasing on (− 3 , 0)∪(0, 3)
(d) f (−3) = − 2 9 is local minimum of f and^ f^ (3) =^
2 9 is local maximum of f (e) f is concave downward on (−∞, −
- and f is concave upward on (−
(f) inflection points (−
√^15 183 ) and (
√^15 183
10 8 × 8 × 4
11 (a) 2 3
x^3 + ln |x| + 2 cos x + C (b) 2 15
4
(2x^3 + 8)^5 + C
12 y′^ = 6 cos 3x, y′′^ = −18 sin 3x
13 y 2 = 4 3
x^3 + C
