Calculus (MT-114)
Instructor: Dr. Faqiha Sultan
NED-University of Engineering
& Technology
Assignment
Due on: 15-01-2020
Instructions:
•
This is a group assignment. Groups of maximum 3 students can be formed.
•
Names and Roll Nos. of all members should be mentioned on the cover page.
•
No assignment would be accepted after the deadline.
Question 1: The accompanying schematic diagram represents an electrical circuit consisting of
an electromotive force that produces a voltage V , a resistor with resistance R, and an inductor
with inductance L. It is shown in electrical circuit theory that if the voltage is first applied at time
t = 0, then the current I flowing through the circuit at time t is given by
What is the effect on the current at a fixed time t if the resistance approaches 0 (i.e., R→0+)?
Question 2: In each part, use the graph of y = f(x) in the accompanying figure to find the
requested information.
(a) Find the intervals on which f is
increasing.
(b) Find the intervals on which f is
decreasing.
(c) Find the open intervals on which f is
concave up.
(d) Find the open intervals on which f is
concave down.
(e) Find all values of x at which f has an
inflection point.
Question 3:
(a) Use both the first and second derivative tests to show that f(x) = x3 − 3x + 3 has a relative
minimum at x = 1 and a relative maximum at x = −1.
(b) Use both the first and second derivative tests to show that f(x) = sin2x has a relative minimum
at x = 0.
(c) Use both the first and second derivative tests to show that g(x) = tan2x has a relative minimum
at x = 0.
(d) Give an informal verbal argument to explain without calculus why the functions in parts (b)
and (c) have relative minima at x = 0.
(e) Show that both of the functions f(x) = (x − 1)4 and g(x) = x3 − 3x2 + 3x − 2 have stationary
points at x = 1.
(f) What does the second derivative test tell you about the nature of these stationary points?
(g) What does the first derivative test tell you about the nature of these stationary points?
Question 4: Use the graphs 1 and 2 to find x-coordinates of the relative extrema and absolute
extrema of f on [0, 7].
Question 5: An open box is to be made from a 16-inch by 30-inch piece of cardboard by cutting
out squares of equal size from the four corners and bending up the sides (Figure given below).
What size should the squares be to obtain a box with the largest volume?
Question 6: The accompanying figure shows an offshore oil
well located at a point W that is 5 km from the closest point
A on a straight shoreline. Oil is to be piped from W to a shore
point B that is 8 km from A by piping it on a straight line
under water from W to some shore point P between A and B
and then on to B via pipe along the shoreline. If the cost of
laying pipe is $1,000,000/km under water and $500,000/km
over land, where should the point P be located to minimize
the cost of laying the pipe?
Question 7: Find a function f such that f ’’(x) = x + cos x and such that f(0) = 1 and f ’(0) = 2.
[Hint: Integrate both sides of the equation twice.]
Question 8: At each point (x, y) on the curve, y satisfies the condition d2y/dx2 = 6x; the line y =
5 − 3x is tangent to the curve at the point where x = 1. Find an equation of the curve that satisfies
the above conditions.
Question 9: The speed of sound in air at 0◦C (or 273 K on the Kelvin scale) is 1087 ft/s, but the
speed v increases as the temperature T rises. Experimentation has shown that the rate of change
of v with respect to T is
where v is in feet per second and T is in kelvins (K). Find a formula that expresses v as a function
of T.
Question 10: Suppose that a uniform metal rod 50 cm long is insulated laterally, and the
temperatures at the exposed ends are maintained at 25◦C and 85◦C, respectively. Assume that an
x-axis is chosen as in the accompanying figure and that the temperature T (x) satisfies the
equation
Find T (x) for 0 ≤ x ≤ 50.
Question 11: Find the volume of the solid generated when the region enclosed by f (x ) = x ,
y = 2, and x = 0 is revolved about the y-axis.
Question 12: Find the volume of the solid that
results when the shaded region is revolved about
the indicated axis. Also sketch the solid.
Question 13: Obtain the third degree Taylor polynomial of f (x ) = x at x = 25 and use it to
approximate the value of
26.5 .
Question 14: Find the fourth degree Taylor polynomial of f (x ) = e
estimate e −0.1 .
x
2
at x = 0 and use it to
Question 15-16: Use sigma notation to write the Taylor series about x = x0 for the function.
Question 17-20: A particle moves along an s-axis with position function s = s(t) and velocity
function v(t) = s’(t). Use the given information to find s(t).
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