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The practice midterm exam for mathematics 2210, a university-level course in vector calculus, taught by mike wills during the autumn 2006 semester. The exam consists of 8 problems covering topics such as vector multiplication, dot product, angle between vectors, plane equations, and vector functions. Students are encouraged to time themselves and note that the actual midterm questions may differ in content but maintain a similar level of difficulty.
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Mathematics 2210 Name: Autumn 2006 Instructor: Mike Wills
Practice Midterm 1
Instructions: This is a practice midterm. Try to time yourself on it. Be aware that the questions on the actual midterm may be different in content from the questions here, but the level of difficulty will be similar. Problem #1 (5 points):
Let v = (2, 1 , 3) and w = (1, 4 , 5). Compute v × w.
Let a, b ∈ Rn. Let θ ∈ [0, π] be the angle between a and b. Show that a · b = |a||b| cos θ. (Hint: Consider a triangle with sides of length α, β, and γ. Let θ be the interior angle between the sides of length α and β. Then the cosine formula says that α^2 + β^2 = γ^2 + 2αβ cos θ.)
Sketch the surface in R^3 defined by the equation x^2 + y
2 4 +^
z^2 9 = 1.
Sketch the surface in R^3 which is given in spherical coordinates by φ = π 4.
Let f (t) = 1 + t if t 6 = 0, and define f (0) = 55. Let r(t) = (f (t), 3 t). Using the precise definition of a limit, show that lim t→ 0 r(t) = (1, 0).
Let a, b, c ∈ R^3. Show that a × (b + c) = (a × b) + (a × c).
