Calculus Answers

A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time t = 0 and allowed to oscillate . After one third of a secondit is observed that the mass returns to its highest position, which was 4.5 cm above its resting position What is the rate of change of the position of the mass at t = 2.1 seconds?

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