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18.01 Exam 3
Name:
Problem 1 : / 20
Problem 2 : / 30
Problem 3 : / 25
Problem 4 : / 25
Total: / 100
Instructions: Please write your name at the top of every page of the exam. The exam is closed
book, calculators are not allowed, but you are allowed to use your prepared index card. You will
have approximately 50 minutes for this exam. The point value of each problem is written next
to the problem – use your time wisely. Please show all work, unless instructed otherwise. Partial
credit will be given only for work shown.
You may use either pencil or ink. If you have a question, need extra paper, need to use the restroom,
etc., raise your hand.
Name: Problem 1 : / 20
Problem 1 (20 points) A particle moves along the positive xaxis with velocity 5 units/second. How
fast is the particle moving away from the point (0, 3) (which is on the yaxis) when the particle is
7 units away from (0, 3)?
Name: Problem 4, continued
(b)(10 points) For an appropriate choice of a, express the following limit in terms of the Riemann
sum from (a). Use the formula for the antiderivative to compute the limit.
�^ n 1 lim √. n→∞ (^9) n^2 + 16 k^2 k=
Name: Problem 3 : / 25
Problem 3 (25 points) Solve the following separable ordinary differential equation with given initial
condition. (^) � y�(x) = ex +2y, y(0) = 0.
Name: Problem 4, continued
(c)(5 points) π/ 2 sin 7 (θ)dθ −π/ 2
(d)(5 points) � (^2) ln(t^2 ) dt 1 t
(e)(5 points) π/ 4 sin(2t) dt. 0 cos(t)
