University:
Whitman College
Course:
MATH-225 | Calculus III
Academic year:

2021
Views:

163

Pages:

2
Author:

Tianna Sullivan

Exam 1 Sample Questions Be sure to look over your old quizzes and homework as well. 1. Eliminate the parameter to find a Cartesian equation for the curve: x = et , y = 2et . Find the arc length of the curve from t = 1 to t = 2 of the original parametric form, then for the (x, y) form. 2. Find dy/dx and d2 y/dx2 , if x = t3 − 12t and y = t2 − 1 (Hint: We do NOT want to try to convert it first). 3. Convert the polar equation to Cartesian: r = tan(θ) sec(θ) 4. Convert the equation from Cartesian to polar of the form r = f (θ). xy = 4 5. Find the area of the surface obtained by rotating the curve about the x−axis: x = 3t − t3 y = 3t2 0≤t≤1 6. Find the slope of the tangent line to the given polar curve at the point specified by θ: r = 2 − sin(θ) θ= π 3 7. Show that the equation r = a sin(θ) + b cos(θ), where ab 6= 0, represents the equation of a circle. 8. If we have two parallel planes, P1 and P2 : P1 : ax + by + cz + d1 = 0 P2 : ax + by + cz + d2 = 0 Then show that the distance between the planes is √ |d1 − d2 | a2 + b 2 + c 2 9. Graphical problems: Like the back of Quiz 3 (online), prob. 24, 25, 26 on p. 627. 10. Do the lines below intersect? If so, find the point of intersection. If not, find the distance between them. x = 3 + t, y = 2 − t, z = 1 x = 2 + s, y = 1 + 2s, z = 2 − s 11. Find an equation for the surface obtained by rotating the parabola y = x2 about the y−axis. (Hint: In 3-d, if you fix a y−value, what shape should you have in the xz−plane?) 1 12. Find the distance between the origin and the line x = 1 + t y = 2 − t z = −1 + 2t Hint: Take an arbitrary point on the line and form two vectors so that the distance can be found- perhaps with a sine? Hint 2: It is possible to solve this with Calculus- Use it to check your answer. 13. Given the property: a × (b × c) = (a · c)b − (a · b)c If vectors a and b are unit vectors, and c = a × (a × b) • Is c a unit vector? • Is c ⊥ a? • Is c ⊥ b? 14. Assume that a 6= 0. Explain your answer (if “No”, provide an example): (a) If a · b = a · c, does it follow that b = c? (b) If a × b = a × c, does it follow that b = c? (c) If a · b = a · c and a × b = a × c, does it follow that b = c? 2

Calculus III Exam 1 Sample Question Solutions

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