Find the average height of the paraboloid z = x
2 + y
2 over the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
What region R in the xy-plane maximizes the value of RR
R
(4−x
2 −2y
2
) dA? What region R in the xy-plane minimizes
the value of RR
R
(x
2 + y
2 − 9) dA? Give the reason for your answer.
Use a double integral in polar coordinates to find the area of the region common to the interior of the cardioids
r = 1 + cos θ and r = 1 − cos θ.
Evaluate the double integral as iterated integral in two ways : RR
R
xy2 dA; R is the region enclosed by y = 1, y = 2, x = 0,
and y = x
Let 𝑓(𝑥) = ൝ 1 + 2𝑥, 𝑥 ≤ 0 3𝑥 − 2,0 < 𝑥 ≤ 1 2𝑥 ଶ − 1, 𝑥 > 1 i) Check whether f is discontinuous. If yes, find where? ii) Give a rough sketch of the graph of f.
Determine the concavity , y-intercept , x-intercepts and co-ordinates of vertex of the parabola 𝑓(𝑥) = 5𝑥 2 − 𝑥 − 3.
Find the dimension of rectangular box with the largest possible volume with an open top and one portion to be constructed from 162 sq. inches of cardboard. (Note: The amount of the material used in construction of box is xy+2xz+2yz=162)
A stone is dropped into a pool of water. The radius of the circular ripple formed increases at 4 𝑚/𝑠. Calculate the rate at which the area of the ripple is increasing when the radius is 6m.
A function is given by the equation 𝑦 = 2𝑥 3 + 4𝑥 2 − 5𝑥 + 10.
i) Obtain an expression for 𝑑𝑦 𝑑𝑥.
ii) Find the gradient of the tangent to the curve at (2, 6).
iii) Determine the equation of this tangent line which passes through (2, 6)
Differentiate the function 𝑓(𝑥) = 5𝑥 2 + 2 using the first principle