Questions from Calculus

Q: Prove that if f is a function of two variables that is

Prove that if f is a function of two variables that is differentiable at (a, b), then f is continuous at (a, b). Hint: Show that

Q: Graph the surface and the tangent plane at the given point.

Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and...

Q: (a). Find the position vector of a particle that has

(a). Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b). Use a computer to graph the path of the particle. a(t) = t i + et j +...

Q: How do you find the tangent vector to a smooth curve at

How do you find the tangent vector to a smooth curve at a point? How do you find the tangent line? The unit tangent vector?

Q: How do you find the length of a space curve given by

How do you find the length of a space curve given by a vector function r(t)?

Q: Reparametrize the curve r(t) = et i + et

Reparametrize the curve r(t) = et i + et sin t j + et cos t k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t.

Q: Find the curvature of the ellipse x = 3 cos t,

Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4).

Q: Find the curvature of the curve y = x4 at the point

Find the curvature of the curve y = x4 at the point (1, 1).

Q: Find an equation of the osculating circle of the curve y =

Find an equation of the osculating circle of the curve y = x4 - x2 at the origin. Graph both the curve and its osculating circle.

Q: A particle moves with position function r(t) = t

A particle moves with position function r(t) = t ln t i + t j + e-t k. Find the velocity, speed, and acceleration of the particle.