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A sample midterm exam for spring math 126 a, b. The exam covers various topics in calculus, including finding the length of a curve, particle motion, normal and osculating planes, and the domain, partial derivatives, and second partial derivative of a function. Students are allowed to use handwritten notes and scientific calculators during the exam.
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Last year’s midterm for Spring MATH 126 A, B
Scientific, but not graphing calculators are OK.
You may use one 8.5 by 11 sheet of handwritten notes.
Problem 1. Consider a particle traveling according to the equations
x(t) = cos^2 t, y(t) = cos t.
Write down and simplify (but do not evaluate) the formula for the length of the curve along which the particle is moving.
Problem 2. Consider a particle whose velocity, at time t ≥ 0, is given by
~v(t) = 〈 − 2 t , − sin t 〉
and whose position at t = 0 is (4, 0).
a. Find the formula for the position of the particle at time t.
b. Find the point at which the particle crosses the y axis.
c. Suppose the acceleration suddenly drops to 0 at the time when the particle crosses the y-axis, so that there are no forces acting on the particle. Find the position of the particle one minute later.
Problem 4. Identify the curve r = 2 sin θ + 2 cos θ
by finding a Cartesian equation for the curve. Give a verbal description of what that curve is.
Problem 5. Consider the function of two variables
f (x, y) =
√ 1 + x − y^2.
a. Identify and sketch the domain of f (x, y).
b. Find the partial derivatives fy(x, y) and fx(x, y).
c. Find the second partial derivative fxy(x, y).
d. Find an equation of the tangent plane at the point (1, 1).
