The function is clearly undefined at , but despite all of this the function does have a limit as approaches 0. a) Use MATLAB and ezplot to sketch for , and use the zoom on facility to guess the . You need to include you M-file, output and graph after zooming. Compute the above limit, and verify that your guess is correct. [Hint: use the squeeze law, see page 78 of Edwards and Penney.]
b) The function is not defined at , hence it cannot be continuous
at . Use item a) to define at so that becomes
continuous at . (In other words, show that the discontinuity of
at is removable.)
c) Now we have that exits for the function you defined in b).
Moreover this is continuous at . Find for , and
find . [Hint: For computing , find the right-hand and lefthand
derivatives of at using the definition given in the
statement of Question 5 b).]
d) Use MATLAB and ezplot to sketch for .
MME1 (MATH 1063) Molly 3
QU, SAIBT,2011
3. An aircraft tracking device employs two directional beams emanating from two ground station A and B situated a distance b apart. The angular coordinates , , and the angular velocity are fed into a computer which calculates the motion of the aircraft. For the simple case of horizontal flight of the target in the vertical plane containing the two stations, show that the velocity v of the aircraft is . [Hint: Assume that the aircraft is directly above the first ground station A at time , and after t seconds it is located directly above as shown in Figure 1). The velocity
is .
4. Two flies (twin sisters) are sitting on a spherical balloon while it is being
inflated at a constant rate. Assume that air is being injected into the balloon at
a rate of 5 cubic centimeters per second (cm3/s), and that balloon has no air in
it to begin with. Further assume that one sister is situated at the North Pole and
the other on the equator.
a) Draw a picture showing an instant of the balloon with the two sisters on it,
complete with labels, including the radius r of the balloon and the
MME1 (MATH 1063) Molly 4
QU, SAIBT,2011
distance s between the two sisters. Note that s is the shortest distance
measured on the surface of the balloon.
b) How fast are the two sisters parting company as a function of the radius r?
c) How fast when r = 1 cm?
d) How fast when r = 0.1 cm?
e) How fast when r = 0.01 cm?
f) How fast are they separating initially?
5. In an electric circuit, for time (t in seconds) and input voltage is
applied, such that
By inspection of , you can see that is continuous at the point ,
and . The charge (in Coulombs) produced by the voltage is
as follow:
YOU MAY ASSUME ALL OF THE ABOVE INFORMATION.
a) Use MATLAB to graph for .
b) Given a function f : R R, the left-hand and right-hand derivatives of f
at a are defined by
, ,
provided these limits exist. The derivative exists if and only if both
limits and exist and coincide. In the latter case,
. Compute the left-hand and right-hand derivatives
of at point and . Use these limits to determine whether
or not and exist.
c) Can the result of item b) be obtained by inspecting the graph of ?
Justify your answer.
MME1 (MATH 1063) Molly 5
QU, SAIBT,2011
d) Compute the right-hand and left-hand limits of at point and
. Use these limits to determine whether or not is
continuous at point and .
6. Suppose that , are acute angles. Assume that and .
Prove that . [Hint: Compute .]
7. Consider the following simple mechanism consisting of two levels AP, PB
loosely hinged at the "elbow" P so that it can bend to the right. The end A is
hinged to a fixed point 8 cm above an origin O. The level AP has length 5 cm,
and the level BP has length 12 cm. The end B is able to slide along the x-axis.
Let be the angle at the elbow P as shown. Clearly P must travel an arc of a
circle of radius 5 cm centered at A, but it is restricted by the end B having to
lie on the x-axis.
Figure 2: Mechanic elbow
To ensure you comprehended the movement of this mechanism, the following picture shows nine different positions as well as the arc traced out by P. Examine carefully each instant as B moves from left to right. MME1 (MATH 1063) Molly 6
QU, SAIBT,2011
Figure 3: The movement of mechanic elbow
a) Suppose the end B is at the position (x, 0). Assume that x is positive at the
right of O, and negative at the left of O. Show that the angle is given
by the relationship . [Hint: Use the law of cosines.]
Using this formula only, find the domain of possible x value. Using the
formula and the figures of the mechanism, carefully explain why the range
is .
b) Use simple algebra or geometry to find the value of x in each of the three
special situations:
1) When is a right-angle and .
2) When P is vertically below A.
3) When P is as high as possible.
c) Let the other angle be as shown in the first figure of this question. Use
trigonometric identities to show that, at the instant in part 1) of item b) .