MATV201 Tutorial 1
1. Which of the points 𝐴 = (23,92,48), 𝐵 = (−60,0,0), 𝐶 = (60,1, −92) is closest to the 𝑦𝑧plane? Which one lies on the 𝑥𝑧-plane? Which one is furthest from the 𝑥𝑦-plane?
2. A sphere has center (2,3,7) and lowest point (2,3, −1). What is the highest point on the
sphere?
3. Find an equation of the sphere with radius 5 and centered at (1,0,0).
4. In each of the cases below, sketch a graph of the surface and briefly describe it in words.
(a) 𝑧 = 3
(b) 𝑧 = 𝑥2 + 𝑦2 + 4
(c) 𝑧 = 𝑦2
(d) 𝑥2 + 𝑦2 = 4
(e) 𝑥2 + 𝑦2 + 𝑧2 = 9
(f) 𝑧 = 7 − 𝑥2 − 𝑦2
(g) 2𝑥 + 4𝑦 + 3𝑧 = 12
(h) 𝑥2 + 𝑧2 = 4
5. Find the equation of the cylinder with radius √7 with its axis along the 𝑦 −axis.
6. Are the following statements true or false? Give a reason for your answers.
(a) The plane 𝑥 + 2𝑦 − 3𝑧 = 1 passes through the origin.
(b) The sphere 𝑥2 + 𝑦2 + 𝑧2 = 10 intersects the plane 𝑥 = 10.
7. In the following cases, sketch a contour diagram for the function with at least three labelled
level curves.
(a) 𝑓(𝑥, 𝑦) = 3𝑥 + 3𝑦
(b) 𝑓(𝑥, 𝑦) = −𝑥2 − 𝑦2 + 1
(c) 𝑓(𝑥, 𝑦) = 𝑥𝑦
8. Find an equation of the level curve of 𝑓(𝑥, 𝑦) = 7𝑥2𝑦2 − 2𝑥 + 4𝑦2 − 10 that goes through
the point (2,4).
9. Are the following statements true or false? Be sure to give reasons for your answers.
(a) If the level curves of 𝑧 = 𝑔(𝑥, 𝑦) are getting closer together in a certain direction, then 𝑓
is increasing in that direction.
(b) If all the level curves of 𝑧 = 𝑓(𝑥, 𝑦) are parallel lines, then the graph of 𝑓 is a plane.