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The measurement of an edge of a cube is found to be 15 cm with a possible error of 0.01 cm. Find the approximate error in computing a.) the volume. b.) the area of one of the faces.
The graph below is made of three line segments:
-1 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
y
x
f(x)
g(x)
h(x)
The segments correspond to the following three functions:
f(x) = x − 2, g(x) = p
4 − (x − 6)2 + 2, h(x) = x − 6
Find the total length L of the graph between x = 2 and x = 10.
Q.1: What are the applications of Calculus in engineering?
Q.1: Define differentiation and integration with example. What are the differences between them?
Q.3: Integrate the following functions with respect to x:
Sin3x, , xy, , 10 .
Q.4: Describe geometrical meaning of indefinite integral. Write down
some properties of indefinite integral.
The drawing below shows a square with side a. A straight line intersects the square and encloses
an area A. The heights x and y on the left and right side (in a distance d from the square) of
the intersecting line can be varied. Assuming that x y and x; y a, nd an expression for
the enclosed area A(x; y) with respect to x and y.
4.Find the area bounded by the cardiode (i) r=a(1+cos theta) , (ii) r=a( 1-cos theta) .
If l,m, be the portions of the axes of x and y intercepted by the tangent at any point x, y on the curve (x/a)^2/3+
(y/b)^2/3=1 , show that (l)^2/(a)^2 + (m)^2/(b)^2 =1
Consider the function f(x)=7(x−4)^2/3
f(x)=7(x−4)2/3. For this function there are two important intervals: (−∞,A)
(−∞,A) and (A,∞)
(A,∞) where A
A is a critical number.
5. A model that describes the population P(t) of a fishery in which harvesting takes place at a constant rate is given by d = kph where k and h are positive constants. dP dt
(a) Solve the DE subject to P(0) = Po
(b) Describe the behavior of the population P(t) for increasing time in the three cases for Po >/, Po = /2 and 0 < Po < 1/1
(c) Use the results from part (b) to determine whether the fish population will ever go extinct in finite time, that is, whether there exists a time T> 0 such that P(T) = 0. If the population goes extinct, then find T.
Find the mass and center of gravity of the cylindrical solid that has density δ(x, y, z) = h − z and is enclosed by
x
2 + y
2 = a
2
, z = 0, and z = h
Find the centroid of the region enclosed by the cardioid r = a(1 + sin θ)
