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calculus assignment, need help with all questions, excluding maple questions

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xxxxx Math 1P97. Assignment 1 (4%). Due date: Monday, May 15, 6:45pm Student’s name:
Total mark: NOTE: Students are expected to complete ALL questions on the assignment. However, only a subset of
questions will be considered for marking. Marks will be deducted for incomplete assignments. Show all
your work.
1. Simplify the following expressions:
a) 2(x2 + 4x − 12) + x2 + 11x + 30
=
x+6 b) 3x2 + 2xy − y 2
xy − 3y 2
=
− 2 + 10 · 2
(3y + x)(x + y)
x − 9y 2 c) 1 y−x
1 x 
y−
x − 2
y
x
=
y 

1 x−y
1
2
x−
y − 2
x
y  2 2. Find the domain of f (x):
a) f (x) = √
3 x−1
;
| +1|
x2 √
b) f (x) = x 16 − x2 ; 2 24−x
c) f (x) =
.
x−2 3. Find the rules for the composite functions (g ◦ f )(x) and (f ◦ g)(x);
a) If f (x) = 5x and g(x) = x5 + 5x + 5; b) If f (x) = 4 − 4x + x2 and g(x) = c) If f (x) = √ x2
,
x−4 x2 + 1 and g(x) = ln x. 4. (Maple Question) An office building worth $16, 166, 050 million when completed in 1980 was depreciated linearly over 50 years to $10, 000, 000.
a) Express the book value of the building V (n) as a function of year starting from 1980: n =
0, V (0) = 16, 166, 050. b) Build a graph of the function V = V (x), x ∈ [0, 50] and find the book value of the building in
2017 using Maple. Attach the Maple worksheet. c) Find the rate at which the building is being depreciated each year. 5. Find the indicated limits, if they exist.
a) e3x (x + 4)
=
x→0 x2 + 6x − 16
lim 2x2 − 3x − 2
b) lim 2
=
x→2 x + x − 6 c) d) 1 − 3x2 + 5x5
√
=
x→∞
1 + x10
lim √ lim
x→∞ e) x2 + 5x −
7 √ x2 − 9x = x4 + 10x3 + 16x2
=
x→−2
x4 − 16
lim 6. Let f =  4 − x2 ,
x3 − 2x2 + a, if x ≤ 2
Find the value of a that will make f continuous on (−∞, ∞).
if x > 2 8. (Commissions, problem 68) The base monthly salary of a salesman working on commission is $26, 000.
For each $50, 000 of sales beyond $100, 000, he is paid a $1000 commission.
a) Sketch a graph y = f (x), x ∈ [100, 300] showing his earnings y as a function of the level of his
sales x (in thousands). b) Determine the values of x for which the function f is discontinuous and find their one-sided
limits. 9. (Energy expended by a fish, problem 72) Suppose a fish swimming a distance of L ft at a speed of
Ï… ft/sec relative to the water and against a current flowing at the rate of u ft/sec (u < Ï…) expends
a total energy given by
aLÏ… 3
E(Ï…) =
,
υ−u
where E is measured in foot-pounds (ft-lb) and a is a constant.
a) Evaluate lim E(Ï…) and interpret your result.
υ→u+ b) Evaluate lim E(υ) and interpret your result.
υ→∞

 


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