Calculus Answers

Let D = {(x, y) ∈ R² : x > 0, 0 < y < x3}. Define

f(x, y) = (0, (x, y)not belong to D

1, (x, y) ∈ D.

a) Approaching (0, 0) along the line y = mx for each real number m and the y-axis, prove that

lim(x,y)→(0,0) f(x, y) exists and compute the limit.

b) Argue whether f is continuous at (0, 0).

Consider the shuttle carrying the astronauts is dropped from 100ft. The position 𝑠 of the shuttle is given by the following function as a function of time: 𝑠(𝑡) = −16𝑡 2 + 100

Consider the interval [0, c] where 0 is the time when the shuttle drops, and c is the time when the shuttle hits the surface of the Earth. Find the height or position of the shuttle when time is: 0, 0.5, 1, 1.5, 2, and 2.5 seconds. What is the domain of this function? Also, sketch the graph of the position s. 

Show that the two curves (e', e², 1-e) and (1-theta , cos theta , sin theta ) intersect at the point (1, 1, 0). What is the angle between their tangents at that point?