QUESTION
Engineering Computations One:
Differential and Integral Calculus
Assignment Two
Due Thursday 5 April 2012
Please ensure that all your working is clearly set out, all pages are stapled
together, and that your name along with the course name appears on the front
page of the assignment. Please place completed assignments in the assignment
box in WT level 1 by 5pm on the due date.
Questions:
1. Evaluate each of the following limits or explain why it does not exist:
lim
lim
lim
x→2
2x
h→0
e
t→2
t
2
+1
x
2
+6x−4
5+h
2
−4
t
2
h
+4
−e
5
lim
lim
lim
x→2
x
4
−16
x−2
x→2
|x−2|
x→0
x−2
|x| e
sin(π/x)
.
2. The current I at time t seconds in a series circuit containing only a resistor
with resistance 10 ohm, an inductor with inductance 0.5henry,anda
steady 12 volt battery connected at time t = 0 is given by the formula
I =
6
5
1 − e
−20t
(this is shown on page 84 of the course manual). Briefly
explain what happens to the current for t ≥ 0.
3. Sketch the graph of a function that satisfies all of the given conditions:
(a) f
(−1) = 0, f
(1) does not exist, f
(x) < 0if|x| < 1, f
(x) > 0if
|x| > 1, f(−1) = 4, f(1) = 0, f
(x) < 0ifx =1.
(b) Domain g =(0, ∞), lim
x→0
+ g(x)=−∞, lim
x→∞
g(x)=0,g
(1) =
0, g
(3) = 0, g
(x) < 0if1<x<3, g
(x) < 0ifx<2orx>4,
g
(x) > 0if2<x<4.
4. At what point(s) on the curve y =2(x − cos x) is the tangent parallel to
the line 3x−y = 5? Illustrate by drawing a rough sketch of the curve, the
line, and the tangent(s).
5. Suppose that f and g are differentiable functions and that F is the function
given by F(x)=f(x)g(x).
(a) Show that F
= f
g +2f
g
+ fg
1
.
(b) Find similar formulas for F
.
6. Find when the function f(x)=
2x
2
and F
+x−1
x
2
+x−2
(4)
is increasing and decreasing.
7. Use calculus to determine whether the graph of f(x)=
cos x
x
is concave
upward or concave downward at x = π.
8. Suppose the position s of a particle at time t is given by s(t)=t tan t for
0 ≤ t ≤
π
3
.
(a) Find the velocity when t =
π
4
.
(b) Is the particle accelerating or decelerating when t =
.
The fact that
d
dx
sec
2
x =
2sinx
cos
3
x
might be useful.
2
π
4
SOLUTION
(a)
(b)
(c)
(d) Left-hand limit
Right-hand limit
Therefore,
Therefore, the limit does not exist.
(e)
(f)
Left-hand limit
Right-hand limit
Therefore,
2.
For
For
Therefore,
,
4.
Given,
Therefore, we have
The tangent is parallel to the line
Its slope
Therefore,
Therefore,
The point is
5.
(a) Given,
By Leibnitz’ rule,
Therefore,
(b)
6. Given,
For critical points,
In the interval
In the interval
Therefore, the function is increasing on and decreasing on .
7. Given,
Therefore,
At the point
Therefore, the function is concave upwards.
- Given,
(a) The velocity is given by
When,
The velocity is
(b) The acceleration is given by
When,
Therefore, the particle is accelerating.
KC33
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