Calculus Answers

Use the implicit differentiation formula to find dy/dx for the equation arctan(x^2 y) = x + xy^2

Find the linearization of the function f(x,y) = ye^x + 2x^2 y at the point (0,2) and use the result to approximate f (0.1, 1.9) to three decimal places.

If y=sin2ax+cosax,prove that yn=a^n{1+(-1) sin2ax}^1/2

Let the displacement function of a moving object is S(t)= 5t^3-3t^2+6t. What is the function for the velocity of the object at time t.

Let f(x)=x^3-6x^2+3x+10, then choose the set of correct options regarding f(x).

• If x element of (-\infinity, 1] union (2, 5) x∈(−∞,1]∪(2,5), then f(x) is negative.

• If x element of [-2, 2] union (5, \infinity)x∈[−2,2]∪(5,∞), then f(x) is positive.

• If x element of [0, 1] union (10, \infinity)x∈[0,1]∪(10,∞), then f(x) is positive.

• If x element of [2, 3] union (5, \infinity) x∈[2,3]∪(5,∞), then f(x) is positive.

• If x element of (-1, 2) union (5, \infinity) x∈(−1,2)∪(5,∞), then f(x) is positive.

(i) At what rate is the area of a rectangle changing if its length is 15 𝑓𝑡 and increasing at 3𝑓𝑡/𝑠𝑒𝑐 while its width is 6𝑓𝑡 and increasing at 2𝑓𝑡/𝑠𝑒𝑐? (ii) The pressure, 𝑝, and the volume, 𝑉, are connected by the equation 𝑝𝑉 1.4 = 𝐾. Determine the approximate percentage error in 𝑘 when the pressure is increased by 4% and the volume is decreased by 1.5%.

An offshore oil well is located in the ocean at a point 𝑾, which is 5 𝑚𝑖𝑙𝑒𝑠 from the closest shore-point 𝑨 on a straight shoreline. The oil is to be piped to a shore-point 𝑩 that is 8 𝑚𝑖𝑙𝑒𝑠 from 𝑨 by piping it on a straight line underwater from 𝑾 to some shore-point 𝑷 between 𝑨 and 𝑩 and then on to 𝑩 via a pipe along the shoreline. If the cost of laying pipe is $100,000.00 per mile under water and $75,000.00 per mile over land, where should point 𝑷 be located to minimize the cost of laying the pipe?

Find all 4 of the second order partial derivatives for the function f(x,y)=e^-xy +x^2y cos(x) -y^2

State the range of function f(x)=(4+x³)² with the given domain -1<x≤1 where x∈ℝ.