Download Calculus I Exam Questions and Answers (Winter 2012, Dawson College, Mathematics) and more Exams Calculus in PDF only on Docsity!
Final Examination: Dawson College: Department of Mathematics: Winter 2012
201-NYA-05: Calculus I (Regular - Social Science)
Question 1. If the limit exists, find its value; otherwise explain why the limit doesnโt exist.
a. (3 marks)
lim xโ 9
x โ 3
Answer: DNE
b. (3 marks)
lim xโโโ
3 x^3 โ x + 1
2 โ 5 x^3
Answer: โ^3 5
c. (3 marks)
lim xโโ 2
x
2 โ 4
x^4 + 2 x^3
Answer:
1 2
Question 2. (5 marks) For which values of x is the following function continuous? Clearly explain your reasoning.
f (x) =
3 xโ 2 x^2 โ 1
if x < โ 2
2 if x = โ 2
x
4
3
Answer: R{โ 2 }
Question 3. Given
f (x) = x
2 โ 2 x + 1
a. (4 marks) Find the derivative of f (x) using the definition of the derivative as a limit.
Answer: f โฒ(x) = 2 x โ 2
b. (3 marks) Find the tangent to f (x) at x = 2.
Answer: y = 2 x โ 3
c. (3 marks) Sketch the graph of f (x) and its tangent at x = 2.
Question 4. (4 marks) Find the absolute maximum value and the absolute minimum value of the given function
f (x) =
x โ x^2 + 1
on [โ 1 , 1 ].
Answer: abs. min. f (โ 1 ) = โโ^1 2
, abs. max. f ( 1 ) = โ^1 2
Question 5. Find the derivative of the following functions:
a. (5 marks)
f (x) = arcsin e^2 x
Answer: f โฒ(x) = โ^2 e^2 x 1 โe^4 x
b. (5 marks)
f (x) =
(ln x + x^2 )^2
sin x
Answer: f
โฒ (x) =
(ln^ x+x^2 )[(^2 x +^4 x)sin^ xโ(ln^ x+x^2 )cos^ x] sin^2 x
c. (5 marks)
f (x) = (x
2
Answer: f โฒ(x) = 1 + 2 x arctan x
Question 6. (5 marks) Using logarithmic differentiation, find the derivative of the function
f (x) = (x
3
12 3
2 x^2 + 5 x(x tan x)
3 .
Do not simplify (expand) your answer.
Answer: f โฒ(x) = (x^3 + 1 )^12
2 x^2 + 5 x(x tan x)^3
[
36 x^2 x^3 + 1
4 x+ 5 6 x^2 + 15 x
3 x +^
2 sec^3 x tan x
]
Question 7. (5 marks) Find an equation of the tangent line to the curve x^2 y^3 โ y^2 + xy โ 1 = 0 at the point ( 1 , 1 ).
Answer: y = โ
3 2 x^ +^
5 2
Question 8.(5 marks) The quantity demanded each month of the IWW Little Red Songbook, is related to the price
per book and given by
p = โ 0. 0000002 x^3 + 6
where p denotes the unit price in dollars and x is the number of books demanded. If the rate of decrease in price is 1$
per month then what is the rate of change of demand of the book when the demand is 50?
Answer: the demand is increasing at a rate of 667 per month
Question 9. The quantity demanded each month of the IWW Little Red Songbook 38th Edition, is related to the price
per book. The equation
P(x) = โ 0. 0000002 x
4
3
- 4 x โ 200 0 โค x โค 310
where x is the number of books demanded, gives the profit function. The total monthly cost (in dollars) for pressing
and packaging x copies of this classic book is given by
C(x) = 200 + 2 x โ 0. 0003 x
3 .
Question 12.(4 marks) Find the horizontal asymptote(s) and vertical asymptote(s) (if any) of the following function.
f (x) =
x^2 โ 2 x
x^2 โ 1
Answer: x = 1 , x = โ 1 , y = 1
Question 13. (4 marks) Evaluate the following integral.
โซ ( โ x +
โ (^3) x โ
x
x
dx
Answer:
2 x^3 /^2 3 +^
3 x^2 /^3 2 โ^ 3 ln^ |x|^ +^3 e
x (^) + c
Question 14. (5 marks) A rectangular box is to have a square base and its volume must be 2m^3. The material for the
base of the box is 200$ per square meter and the material of the side and the top 150$ per square meter. If the cost of
shipping is determined by the size of the base of the box 500$ per square meter then determine the dimensions of the
box to minimise the cost of construction of the box and shipping.
Answer: base=
3
12 17 , height=^
2 ( 3
โ 12 17
) 2
