1223-UCONN-MATH-2110Q-SEC100-6938 Exam 1 Version
B
Daham Jayasena
TOTAL POINTS
QUESTION 1
the chain rule so this is TRUE.
Multiple Answer 16 pts 1.3 Tangent Plane Approx
1.1 Directional Derivative Possibilities 2 / - 0 pts Correct! Plugging $$(3/4,2)$$ into the
4 tangent plane gives the approximation
- 0 pts Correct! Since $$||\vec{\nabla}f||=5$$ $$f(3/4,2)=\ln(3/2)\approx 1/2$$.
the max directional derivative is 5 and the min is ✓ - 4 pts Incorrect. Plugging $$(3/4,2)$$ into the
-5. tangent plane gives the approximation
✓ - 1 pts (Chose -10) Since $$||\vec{\nabla}f||=5$$ $$f(3/4,2)=\ln(3/2)\approx 1/2$$.
the max directional derivative is 5 and the min is -5,
1.4 Lagrange Equations
so you should choose $$-2$$ and $$4$$ only.
✓ - 0 pts Correct! The two equations in the system
- 1 pts (Did not choose -2) Since
are $$\langle 2x, 2y, 2z \rangle = \lambda \langle 2,
$$||\vec{\nabla}f||=5$$ the max directional
3, 2z \rangle$$ and $$2x+3y+z^2=7$$.
derivative is 5 and the min is -5, so you should
- 1 pts You chose $$\langle x^2, y^2, z^2 \rangle
choose $$-2$$ and $$4$$ only.
= \lambda \langle 2, 3, 2z \rangle$$ but this is not
- 1 pts (Did not choose 4) Since
the correct gradient of f.
$$||\vec{\nabla}f||=5$$ the max directional
- 1 pts You didn't choose $$\langle 2x, 2y, 2z
derivative is 5 and the min is -5, so you should
\rangle = \lambda \langle 2, 3, 2z \rangle$$ but
choose $$-2$$ and $$4$$ only.
this is the expression $$\vec{\nabla}f=\lambda
✓ - 1 pts (Chose 8) Since $$||\vec{\nabla}f||=5$$
\vec{\nabla}g$$ so it should have been chosen.
the max directional derivative is 5 and the min is -5,
- 1 pts You didn't choose $$2x+3y+z^2=7$$ but
so you should choose $$-2$$ and $$4$$ only.
should have.
1.2 Chain Rule - 1 pts You chose $x^2+y^2+z^2$ but this isn't
✓ - 0 pts Correct! This is the correct statement of the an eqaution!
chain rule.
QUESTION 2
- 4 pts Incorrect. This is the correct statement of
, 2 Matching with above).
✓ - 0 pts Correct - 1 pts Incorrect point
- 2 pts Correct choices are A, B, C, D - 1 pts Incorrect equation.
- 4 pts Correct choices are A, B, C, D - 1 pts Incorrect simplification. You should get
- 6 pts Correct choices are A, B, C, D the equation of the plane is $$-20x+6y+5z=-31$$.
- 8 pts Correct choices are A, B, C, D - 2 pts incorrect formula for the plane.
QUESTION 3 QUESTION 4
Planes 8 pts Critical Points 8 pts
3.1 Vector Orthogonal to Both 4.1 Which are CPs?
✓ - 0 pts Correct! The cross product between those ✓ - 0 pts Correct! The critical points are $$(3,2)$$
two vectors, which gives the vector orthogonal to and $$(1,-4)$$.
both is $$\langle -20, 6, 5 \rangle$$. - 2 pts Did not select both critical points. You
- 1 pts You forgot the extra $$-1$$ on the should have both $$(3,2)$$ and $$(1,-4)$$ since
$$\vec{j}$$ component. You should have gotten $$g_x=g_y=0$$ for both of those.
$$\langle -20, -6, 5 \rangle$$. - 2 pts You selected extra points. You should
- 1 pts Incorrect $$\vec{i}$$ component. You only have both $$(3,2)$$ and $$(1,-4)$$ since
should have gotten $$\langle -20, 6, 5 \rangle$$. $$g_x=g_y=0$$ for both of those.
- 1 pts Incorrect $$\vec{j}$$ component. You
should have gotten $$\langle -20, 6, 5 \rangle$$. 4.2 Classifying CPs
- 1 pts Incorrect $$\vec{k}$$ component. You ✓ - 0 pts Correct! (3,2) is a local min and (1,-4) is a
should have gotten $$\langle -20, 6, 5 \rangle$$. saddle.
- 4 pts Blank or incorrect. You should have - 0 pts Correct classification of wrong points.
gotten $$\langle -20, 6, 5 \rangle$$. The "correct" classifications are:
- 0 pts Technically correct. $$\vec{0}$$ is (0,4) inconclusive
orthogonal to all vectors. (3,2) min
(-2,-3) inconclusive
3.2 Equation of Plane (-4,-1) min
✓ - 0 pts Correct! The equation of the plane is $$- (1,-4) saddle.
20x+6y+5z=-31$$. - 2 pts Incorrect classification of a point. Note:
- 4 pts Incorrect or blank. don't take off point for classifying extra points,
- 0 pts Consistent work with error above. only for wrong classification. The "correct"
- 1 pts Incorrect normal vector (inconsistent classifications are:
B
Daham Jayasena
TOTAL POINTS
QUESTION 1
the chain rule so this is TRUE.
Multiple Answer 16 pts 1.3 Tangent Plane Approx
1.1 Directional Derivative Possibilities 2 / - 0 pts Correct! Plugging $$(3/4,2)$$ into the
4 tangent plane gives the approximation
- 0 pts Correct! Since $$||\vec{\nabla}f||=5$$ $$f(3/4,2)=\ln(3/2)\approx 1/2$$.
the max directional derivative is 5 and the min is ✓ - 4 pts Incorrect. Plugging $$(3/4,2)$$ into the
-5. tangent plane gives the approximation
✓ - 1 pts (Chose -10) Since $$||\vec{\nabla}f||=5$$ $$f(3/4,2)=\ln(3/2)\approx 1/2$$.
the max directional derivative is 5 and the min is -5,
1.4 Lagrange Equations
so you should choose $$-2$$ and $$4$$ only.
✓ - 0 pts Correct! The two equations in the system
- 1 pts (Did not choose -2) Since
are $$\langle 2x, 2y, 2z \rangle = \lambda \langle 2,
$$||\vec{\nabla}f||=5$$ the max directional
3, 2z \rangle$$ and $$2x+3y+z^2=7$$.
derivative is 5 and the min is -5, so you should
- 1 pts You chose $$\langle x^2, y^2, z^2 \rangle
choose $$-2$$ and $$4$$ only.
= \lambda \langle 2, 3, 2z \rangle$$ but this is not
- 1 pts (Did not choose 4) Since
the correct gradient of f.
$$||\vec{\nabla}f||=5$$ the max directional
- 1 pts You didn't choose $$\langle 2x, 2y, 2z
derivative is 5 and the min is -5, so you should
\rangle = \lambda \langle 2, 3, 2z \rangle$$ but
choose $$-2$$ and $$4$$ only.
this is the expression $$\vec{\nabla}f=\lambda
✓ - 1 pts (Chose 8) Since $$||\vec{\nabla}f||=5$$
\vec{\nabla}g$$ so it should have been chosen.
the max directional derivative is 5 and the min is -5,
- 1 pts You didn't choose $$2x+3y+z^2=7$$ but
so you should choose $$-2$$ and $$4$$ only.
should have.
1.2 Chain Rule - 1 pts You chose $x^2+y^2+z^2$ but this isn't
✓ - 0 pts Correct! This is the correct statement of the an eqaution!
chain rule.
QUESTION 2
- 4 pts Incorrect. This is the correct statement of
, 2 Matching with above).
✓ - 0 pts Correct - 1 pts Incorrect point
- 2 pts Correct choices are A, B, C, D - 1 pts Incorrect equation.
- 4 pts Correct choices are A, B, C, D - 1 pts Incorrect simplification. You should get
- 6 pts Correct choices are A, B, C, D the equation of the plane is $$-20x+6y+5z=-31$$.
- 8 pts Correct choices are A, B, C, D - 2 pts incorrect formula for the plane.
QUESTION 3 QUESTION 4
Planes 8 pts Critical Points 8 pts
3.1 Vector Orthogonal to Both 4.1 Which are CPs?
✓ - 0 pts Correct! The cross product between those ✓ - 0 pts Correct! The critical points are $$(3,2)$$
two vectors, which gives the vector orthogonal to and $$(1,-4)$$.
both is $$\langle -20, 6, 5 \rangle$$. - 2 pts Did not select both critical points. You
- 1 pts You forgot the extra $$-1$$ on the should have both $$(3,2)$$ and $$(1,-4)$$ since
$$\vec{j}$$ component. You should have gotten $$g_x=g_y=0$$ for both of those.
$$\langle -20, -6, 5 \rangle$$. - 2 pts You selected extra points. You should
- 1 pts Incorrect $$\vec{i}$$ component. You only have both $$(3,2)$$ and $$(1,-4)$$ since
should have gotten $$\langle -20, 6, 5 \rangle$$. $$g_x=g_y=0$$ for both of those.
- 1 pts Incorrect $$\vec{j}$$ component. You
should have gotten $$\langle -20, 6, 5 \rangle$$. 4.2 Classifying CPs
- 1 pts Incorrect $$\vec{k}$$ component. You ✓ - 0 pts Correct! (3,2) is a local min and (1,-4) is a
should have gotten $$\langle -20, 6, 5 \rangle$$. saddle.
- 4 pts Blank or incorrect. You should have - 0 pts Correct classification of wrong points.
gotten $$\langle -20, 6, 5 \rangle$$. The "correct" classifications are:
- 0 pts Technically correct. $$\vec{0}$$ is (0,4) inconclusive
orthogonal to all vectors. (3,2) min
(-2,-3) inconclusive
3.2 Equation of Plane (-4,-1) min
✓ - 0 pts Correct! The equation of the plane is $$- (1,-4) saddle.
20x+6y+5z=-31$$. - 2 pts Incorrect classification of a point. Note:
- 4 pts Incorrect or blank. don't take off point for classifying extra points,
- 0 pts Consistent work with error above. only for wrong classification. The "correct"
- 1 pts Incorrect normal vector (inconsistent classifications are:
