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Q 2. Evaluate the following iterated integral if region of integration R is bounded by graphs of y = x² and y = 2x.
∫ ∫ (x³+4y)dydx
R
Q 2. Evaluate the following iterated integral if region of integration R is bounded by graphs of y = x² and y = 2x.
∫ ∫ (x³+4y)dydx
R
Q 3. Evaluate the iterated integral by first reversing the order of integration
² 4
∫ ∫ ycosx²dxdy
° y²
Q 1. Find the absolute extrema of the function
f(x,y) = 5+4x-2x²+3y-y² on the triangular region with vertices (0, 0), (2, 2) and (-2, 2).
Question: The sum of three numbers in A.P is 24 and their product is 440.Find the numbers.
Question: For the real valued function, f defined below
Find f^-1(-1)
Where
f(x)=3x^2+7
Question: If f(x)=x/2+3 and g(x)=3/4x-2
then find the value of
5f(-2)-7g(-4)
Q:1:- Prove that the function f(x)=x^3-2x+3 is continuous.
Let A = {1, 2,3} and B={1,2,3,4}, define a binary relation R from A to B as follows
R= {(a, b) E AxB | a b}
(i) Find the order pairs in R.
(ii) Find domain and range of R.
(iii) Draw the Directed graph of R.
(iv) Make the matrix representation of R
Prove that for the sets A, B, and C
(A B) C = (A C) (B C)
Using
(i) Venn diagram
(ii) Membership table
Using the set identities prove the commutative law over intersection.
Q 3. If f(x,y) = x^3/3+4/3y^3-x^2-3x-4y-3
Find relative extrema and saddle points of f.
Q 3. If f(x,y) = x^3/3+4/3y^3-x^2-3x-4y-3
Find relative extrema and saddle points of f.
Q 2. Find equations for the tangent plane and the normal line to the graph of the equation 2e^-x cosy-z = 0 at the point P(0,pi/3,1)
Q 1. Let f(x,y) = x^2 - 4xy?
a. Find the gradient of f at the point P(1,2) .
b. Find directional derivative of f at P(1,2) at in the direction from P(1,2) to Q(2,5)
Let f(x,y) = (x^2-2xy+5y^2/3x^2-4y^2)
a. Find the domain of f(x,y).
b. By using different paths, show that whether the limit of f(x,y) exists at the origin or not.
c. Is the function continuous at origin? Justify your answer.
Write down an equation or equations for each of the following.
a. A curve produced due to the intersection of a circular cylinder and yz-plane. Center of this circular cylinder is x-axis (that is, cylinder is around x-axis) and its radius is one.
b. A curve produced due to the intersection of a sphere, center at origin and radius 3, and xy-plane.
Write down an equation or equations for each of the following.
a. A plane parallel to xy-plane at a distance of 4 units from origin on negative z-axis.
b. A curve produced due to the intersection of a circular cylinder and yz-plane. Center of this circular cylinder is x-axis (that is, cylinder is around x-axis) and its radius is one.
c. A curve produced due to the intersection of a sphere, center at origin and radius 3, and xy-plane.
Let f(x,y) = (x2-2xy+5y2/3x2-4y2)
a. Find the domain of f(x,y).
b. By using different paths, show that whether the limit of f(x,y) exists at the origin or not.
c. Is the function continuous at origin? Justify your answer.
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