Calculus Answers

Use Taylor's formula to find a quadratic approximation of f(x, y) = cos cos y at the origin. Estimate the error in the approximation f|x| <= 0.1 and |y| <= 0.1

Show that (0,0) is a critical point of f(x, y) = x ^ 2 + kxy + y ^ 2 no matter what value the

constant khas. (Hint: Consider two cases: k = 0 and k * 0.)

Let T = f(x, y) be the temperature at the point (r, y) on the circle r = cost, y-inf.0≤ t≤2 and suppose that

ar 8-4y 8y-42

. Find where the maximum and minimum temperatures on the circle occur by examining the derivatives dT> dt and dT/dt

r=4x^ 2 -4xy+4y^ 2 . . Suppose that Find the maximum and minimum values of T7 on the circle

If * (x, y) = 30(y + y ^ 2) represents the population density of a planar region on Earth, where randy are measured in miles, find the number of people in the region bounded by the curves x = y ^ 2, x = 2y - y ^ 2

A space probe in the shape of the sphere x ^ 2 + y ^ 2 + z ^ 2 = 30 enters Earth's atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (z. y. :) on the probe's surface is T(x, y, z) = x - 2y + 5z Find the hottest point on the probe's surface.

When the air resistance is ignored, the horizontal range R of a projectile is given by R(8)= 9 sin 20 where is the constant initial velocity, g is the acceleration due to gravity, and is the angle of elevation or departure. Find the maximum range of projectile.

Use the ideal gas law Pwith volume V in cubic inches (in), temperature 7' in Kelvins (K) and R = 10(inIb / K) to find the rate at which the temperature of a gas is changing when the volume is 200in" and increasing at the rate of 4in/s, while the pressure is 5lb/in² and de creasing at the rate of 11b/in/s.

Find 𝑑𝑦

𝑑𝑥

in each case

(a) 𝑦 = 𝑡𝑎𝑛4(3𝑥)

(3)

(b) 𝑥2𝑦2 + 𝑥𝑠𝑖𝑛𝑦 = 4 use implicit differentiation

find f(x) = x² - 2x -2 find f' (x) using the long method

find f(x) = x2 - 2x -2 find f' (x) using the long method