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Math 230 — Summer 2023

Mastery Assignment 2

Due June 16, 2023 11:59 PM

Instructions:

• Answers to these problems must uploaded to GradeScope using this template (if possible). Otherwise, you must map your answers to the right question in GradeScope at the time of upload. Answers must be organized and legible. Points may be deducted if anything is unclear about the process of your work.

• Points will be deducted for incomplete reasoning and disorganized work (even if your answers are correct).

• If you have any difficulties/questions, you are encouraged to discuss these problems with me.

• Late assignments must go into the Late Assignment Dropbox on Canvas.

1. (5 points) Consider the curve ~r(t) = tˆi + log(t)ˆj.

(a) Compute the curvature at the point (1,0)

(b) For a smooth curve parametrized by ~r(t), the osculating plane at a point ~r(t0) is the plane con-taining both T(t0) and N(t0). The osculating circle is the circle lying in the osculating plane that is tangent to the curve and has the same curvature at the point ~r(t0) (See section 13.4 page 766 for further details and examples) Give the equation of the osculating circle at the point (1,0).

2. (5 points) A roller coaster travels upward along the path ~r(t) = ht, 1/2 t 2, 1/6 t 3i (position here is measured in meters)

(a) Compute the acceleration at t = 4 seconds (include units)

(b) Decompose the acceleration into its tangential and normal components.

3. (5 points) Consider the contour map below.

(a) Sketch and label the following contour plots

• f(2x, y)

• f(x, 2y)

• 2f(x, y)

(b) Sketch the graph of the surface z = f(x, y).

(c) Use the contour plot to estimate

4. (5 points) Compute all first and second order partial derivatives of f(x, y) = exy sin y. Verify that Clairaut’s theorem holds.