Calculus Answers

The product of two numbers is 4 square root of 3. Find the numbers so that the

sum 𝑆 of the square of one and the cube of the other is as small as

possible

derive a reduction formula e^(mx)/x^ndx

Use the exponential series to find an approximation to e-1correct to three significant figures.

Calculate the area of ​​the region R in the xy plane enclosed in the circumference

x² + y²= 4, to the right of line x = 1 and above line x =√3y.

Find the volume of the solid bounded by the planes x = y, x + y + z = 4,

y = 0, z = 0.

Determine the location and values of the absolute maximum and absolute 

minimum for the given function: 

𝑓(𝑥) = (−𝑥 + 2)⁴, 𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑥 ≤ 3

Determine the location and values of the absolute maximum and absolute 

minimum for the given function: 

𝑓(𝑥) = (−𝑥 + 2)

ସ

, 𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑥 ≤ 3

^{(a) Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If}

^{𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥ଷ 𝑦ଶ − 𝑥ଶ 𝑦ଷ + 4𝑥 + 9}

^{(b) A firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars where}

^{𝑄 = 𝑄(𝑥) = −𝑥ଶ + 600𝑥 + 25}

^{i) Over what level of advertising expenditure is the number of units of product sold increasing?}

^{ii) Over what level of advertising expenditure is the number of units of product sold decreasing?} 

Find the value of a and b if

lim x(1+acosx)-bsinx/ (x^3)=1

x→∞