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11/16/2019
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
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MATH 2163, section 61998, Fall 2019
INSTRUCTOR
Sec. 14.5: Gradient and Directional Derivatives
(Calculus3Homework)
Ning Ju
Northern Oklahoma
College
Current Score
QUESTION
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
POINTS
1/1
0/2
–/1
0/1
–/1
–/1
–/1
–/1
–/1
–/1
–/1
–/1
–/2
–/1
–/1
–/2
TOTAL SCORE
1/19
5.3%
Due Date
DECEMBER 14
11:32 PM CST
Assignment Submission & Scoring
Assignment Submission
For this assignment, you submit answers by question parts. The number of submissions remaining for each
question part only changes if you submit or change the answer.
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1/8
11/16/2019
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
Assignment Scoring
Your best submission for each question part is used for your score.
1.
1/1 points
RogaCalcET3 14.5.003.
My Notes
The figure below shows the level curves of a function f(x, y) and a path r(t), traversed in the direction
d
indicated. State whether the derivative
f(r(t)) is positive, negative, or zero at points A-D.
dt
At point A the derivative is zero
.
At point B the derivative is negative
.
At point C the derivative is positive
.
At point D the derivative is zero
.
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2/8
11/16/2019
2.
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
0/2 points
RogaCalcET3 14.5.002.
My Notes
Let f(x, y) = 4exy and r(t) = (t3, 4 + t).
1. Calculate ∇f.
∇f =
2. Calculate r'(t).
r'(t) =
3t2,5
3. Use the Chain Rule for Paths to calculate
d
f(r(t)) =
dt
d
f(r(t)).
dt
4. Write out the composite f(r(t)) as a function of t.
f(r(t)) =
5. Differentiate f(r(t)).
d
f(r(t)) =
dt
Check that the results of (3) and (5) agree.
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11/16/2019
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
–/1 points
3.
RogaCalcET3 14.5.005.
My Notes
f(x, y) = cos(x8 + y)
∇f =
0/1 points
4.
RogaCalcET3 14.5.007.
My Notes
h(x, y, z) = xyz8
∇f =
8xyz7
5.
–/1 points
RogaCalcET3 14.5.010.
Use the Chain Rule to calculate
My Notes
d
f(r(t)).
dt
f(x, y) = 2x − 7y, r(t) = t4, t5 , t = 3
d
f(r(t)) =
dt
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4/8
11/16/2019
6.
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
–/1 points
RogaCalcET3 14.5.011.
Use the Chain Rule to calculate
My Notes
d
f(r(t)).
dt
f(x, y) = x3 − 5xy, r(t) = (cos(3t), sin(3t)), t = 0
d
f(r(t)) =
dt
7.
–/1 points
RogaCalcET3 14.5.015.
Use the Chain Rule to calculate
My Notes
d
f(r(t)).
dt
f(x, y) = 5x − 4xy, r(t) = (t2, t2 − 2t), t = 2
d
f(r(t)) =
dt
8.
–/1 points
RogaCalcET3 14.5.018.
Use the Chain Rule to calculate
My Notes
d
f(c(t)).
dt
f(x, y, z) = x8y7e4z, c(t) = (t2, t3, t − 1), t = 1
d
f(c(t)) =
dt
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5/8
11/16/2019
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
–/1 points
9.
RogaCalcET3 14.5.021.
My Notes
Calculate the directional derivative in the direction of v at the given point. Remember to normalize the
direction vector.
f(x, y) = x2 + y3,
v = 2, 5 ,
P = (5, 2)
Du f(P) =
10.
–/1 points
RogaCalcET3 14.5.025.Tutorial.SA.
My Notes
This question has several parts that must be completed sequentially. If you skip a part of the question, you
will not receive any points for the skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
Calculate the directional derivative in the direction of v at the given point. Remember to normalize the
1
direction vector or use Duf(P) =
∇fP · v.
v
f(x, y) = tan−1(xy),
11.
–/1 points
v = 1, 1 ,
P = (6, 3)
RogaCalcET3 14.5.028.
My Notes
Calculate the directional derivative in the direction of v at the given point. Remember to normalize the
direction vector.
f(x, y, z) = z2 − xy2,
v = −1, 2, 2 ,
P = (4, 5, 8)
Du f(4, 5, 8) =
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11/16/2019
12.
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
–/1 points
RogaCalcET3 14.5.032.
My Notes
Let f(x, y, z) = xy + z4, P = (8, 6, 1). Calculate the directional derivative in the direction pointing to the
origin. Remember to normalize the direction vector.
Du f(8, 6, 1) =
13.
–/2 points
RogaCalcET3 14.5.038.
My Notes
2
Let f(x, y) = xex − y and P = (9, 81).
(a) Calculate
∇fP .
∇fP =
(b) Find the rate of change of f in the direction ∇fP.
(c) Find the rate of change of f in the direction of a vector making an angle of 45° with ∇fP.
14.
–/1 points
RogaCalcET3 14.5.041.
My Notes
Find a vector normal to the surface x2 + y2 − z2 = 40 at P = (4, 5, 1).
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7/8
11/16/2019
15.
Sec. 14.5: Gradient and Directional Derivatives – MATH 2163, section 61998, Fall 2019 | WebAssign
–/1 points
RogaCalcET3 14.5.044.
My Notes
Find an equation of the tangent plane to the surface at the given point.
5×2 + 4y2 + 3z2 = 136,
16.
–/2 points
P = (3, 2, 5)
RogaCalcET3 14.5.046.
My Notes
Find an equation of the tangent plane to the surface at the given point.
x2 + 2z2ey − x = 41,
P=
3, 4,
4
e
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