Math 2205 - SERCalc II Quiz 11 Name: 1. (2 points) Find a vector

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Math 2205 - SERCalc II Quiz 11 Name: 1. (2 points) Find a vector
Math 2205 - SERCalc II
Quiz 11
Name:
1. (2 points) Find a vector equation for the line tangent to the curve given by r(t) = ln(t + 1), tan t, te−t at t = 0.
Recall that such a description has the form l(t) = r(0)+tr0 (0). Write your answer in the form l(t) = hx(t), y(t), z(t)i
where x(t), y(t) and z(t) are all linear functions of t.
2. (2 points) Find the equation of the plane defined by the vectors a = h2, −1, 3i and b = h−1, 0, 1i and containing
the point Po (0, 5, −1). Your answer should be in the form ax + by + cz = d.
3. (2 Dpoints) If F(x,Ey, z) = 2x, 3y + z 2 , cos(2πxy) , find proju F at the point P (1, 1, 2) where u is the unit vector
u = √111 , √−1
, √311 . Do this by evaluating F at P and using that vector in the calculation of the projection
11
vector requested, mathematically F(P ) · u u since u is a unit vector.
4. (4 points) The point of this problem is to find an equation of the plane tangent to the surface S given by
z = f (x, y) = x2 + y 2 at the point Po (1, 2, 5).
(a) Begin by describing S parametrically using x, y as the parameters, that is, write S as
s(x, y) = h
,
i
,
(b) Next find vector equations for two curves on S through the point Po , one holding y constant at y = 2 and the
second holding x constant at x = 1. Call these vectors a(x) and b(y) respectively.
a(x) = s(x, 2) = h
,
,
i
b(y) = s(1, y) = h
,
,
i
(c) Next find tangent vectors to both of the curves determined in (b) by differentiating.
a0 (x) = h
,
,
i
b0 (y) = h
,
,
i
(c) Find a vector orthogonal to the desired plane by calculating a0 (1) × b0 (2).
a0 (1) × b0 (2) = h
,
,
i
(d) Find the equation of the desired plane. Your answer should be in the form ax + by + cz = d.

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