c Dr Oksana Shatalov, Spring 2014
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Spring 2014 Math 251
Week in Review 4
courtesy: Oksana Shatalov
(covering Sections 12.4-12.5)
12.4: Tangent Planes and Differentials
Key Points
• An equation of the tangent plane to the graph of the function z = f (x, y) at the point P (x0 , y0 , f (x0 , y0 )) is
z − f (x0 , y0 ) = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).
• The line through the point P (x0 , y0 , f (x0 , y0 )) parallel to the vector n is perpendicular to the above tangent
plane. This line is called the normal line to the surface z = f (x, y) at P .
• The differential dz (or the total differential): dz =
∂z
∂z
dx +
dy.
∂x
∂y
• ∆z ≈ dz.
• f (a + ∆x, b + ∆y) ≈ f (a, b) + dz(a, b)
1. Sketch the circular paraboloid z = 7 − x2 − y 2 to see why the tangent plane at its vertex
should be horizontal. Confirm that by finding the equation of this tangent plane.
2. Find an equation of the tangent plane to the graph of the function z = ln(3x + 2y) at the
point (−1, 2, 0).
3. Find parametric equations for the normal line to the surface z = ex+y at the point P (1, −2, 1/e).
4. Find an equation of the tangent plane to the surface z = x2 + 3y 3 which is parallel to the
plane 2x + y − z = 1.
5. Where does the plane tangent to the surface z = ex−y at (1, 1, 1) meet the z -axis?
6. Find the differential of the function
v = y ln(cos(x2 + 3))
7. Find the differential of the function
f (x, y, z) =
p
x2 + y 2 + z 2
8. Use differentials to estimate
p
(4.01)2 + (3.98)2 + (2.02)2
c Dr Oksana Shatalov, Spring 2014
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9. The two legs of a right triangle are measured as 5 m and 12 m respectively, with a possible
error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum
error in the calculated value of
(a) the area of the triangle
(b) the length of the hypotenuse.
12.5: The Chain Rule
Key Points
• CASE 1: z = f (x, y), where x = x(t), y = y(t).
∂z dx ∂z dy
dz
=
+
dt
∂x dt
∂y dt
CASE 2: z = f (x, y), where x = x(s, t), y = y(s, t).
∂z ∂x ∂z ∂y
∂z
=
+
∂s
∂x ∂s
∂y ∂s
∂z
∂z ∂x ∂z ∂y
=
+
∂t
∂x ∂t
∂y ∂t
10. If z = y + f (x2 − y 2 ) , where f has continuous partial derivatives, show that
y
∂z
∂z
+x
=x
∂x
∂y
11. Let
w = sin 2x + 3y + y sin x,
where
x = e−t + 3s, y = 5e2t −
Find
s
∂w
∂w
and
.
∂t
∂s
12. Use a chain rule to find the value
13. Find
√
√
du
at s = 1/4 if u = r2 − r tan θ, r = s, θ = πs.
ds
∂z
∂z
and
∂x
∂y
(a) ln(1 + z) + xy 2 + z = 2013
(b) yz 4 − xz 3 = ex
3 yz
14. Two straight roads intersect at right angles. Ben, walking on one of the roads, approaches
the intersection at 0.6m/s and Jill, walking on the other road, approaches the intersection
at 0.5m/s. At what rate is the distance between Ben and Jill changing when Ben is 500m
from the intersection and Jill is 400m from the intersection?
c Dr Oksana Shatalov, Spring 2014
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15. Two sides of a triangle have lengths a = 5m, b = 10m, and the included angle θ = π/6. If
the side a is increasing at a rate 2m/s, and the side b is decreasing at a rate of 1m/s, and
θ remains constant, at what rate is the third side changing? Is it increasing or decreasing?
(Hint: Use the law of cosines.)