## Calculus III

__Task 5: Vectors & Planes__

**Given:**

Plane P_{1} is given by the equation 3x + 4y – 5z = 60

Plane P_{2} is given by the equation 4x + 2y + cz = 0

**Requirements:**

*Your submission must be your original work. No more than a combined total of 30% of the submission and no more than a 10% match to any one individual source can be directly quoted or closely paraphrased from sources, even if cited correctly. Use the Turnitin Originality Report available in Taskstream as a guide for this measure of originality.*

*You must use the rubric to direct the creation of your submission because it provides detailed criteria that will be used to evaluate your work. Each requirement below may be evaluated by more than one rubric aspect. The rubric aspect titles may contain hyperlinks to relevant portions of the course*

- Determine the value of
**C**that makes plane P_{2}perpendicular to plane P_{1}using algebra and/or calculus techniques and justifying all work. - Determine the equations of
**two**non-perpendicular planes (both different from the planes in part A), and show they are not perpendicular using algebra and/or calculus techniques and justifying all work. - Acknowledge sources, using APA-formatted in-text citations and references, for content that is quoted, paraphrased, or summarized.

__Task 6: Gradient & Slope__

**Given:**

*f* is a multivariable function defined by *f(x, y)* = *x ^{3}y* –

*x*where

^{2}y^{2}*x*and

*y*are real variables.

**Requirements:**

*Your submission must be your original work. No more than a combined total of 30% of the submission and no more than a 10% match to any one individual source can be directly quoted or closely paraphrased from sources, even if cited correctly. Use the Turnitin Originality Report available in Taskstream as a guide for this measure of originality.*

*You must use the rubric to direct the creation of your submission because it provides detailed criteria that will be used to evaluate your work. Each requirement below may be evaluated by more than one rubric aspect. The rubric aspect titles may contain hyperlinks to relevant portions of the course.*

- Pick a specific point P
_{0}= (*a, b, f(a, b)*) on the surface*z*=*f(x, y)*, other than (0, 0, 0) or (3, 21, -3402).

*Note: You may use whimsical values for a and* b, *such as the month and day of your birthday. For example, March 21 ^{st} becomes* a=

*3 and*b=

*21, so the 3*.

^{rd}coordinate is f(3, 21) = -3402- Calculate the directional derivative of the vector in the direction of greatest increase of the surface at P
_{0}. Use algebra and/or calculus techniques and justify all work. - Find a direction vector in which the directional derivative of
*f(x, y)*at P_{0}is equal to zero. Use algebra and/or calculus techniques and justify all work. - Acknowledge sources, using APA-formatted in-text citations and references, for content that is quoted, paraphrased, or summarized.

__Task 7: Second Partial Derivative Test__

**Given:**

Multivariable function *f*(*x,y*) = *ky ^{2}* – 2

*y*–

^{3}*kx*+ 2

^{3}*kxy*

**Requirements:**

*Your submission must be your original work. No more than a combined total of 30% of the submission and no more than a 10% match to any one individual source can be directly quoted or closely paraphrased from sources, even if cited correctly. Use the Turnitin Originality Report available in Taskstream as a guide for this measure of originality.*

*You must use the rubric to direct the creation of your submission because it provides detailed criteria that will be used to evaluate your work. Each requirement below may be evaluated by more than one rubric aspect. The rubric aspect titles may contain hyperlinks to relevant portions of the course.*

- The function
*f*(*x,y*)has a critical point at (0,0). Use the second derivative test to demonstrate that for all nonzero values of*k, f*has a saddle point at (0,0). Use algebra and/or calculus techniques to justify all work. - The function
*f*(*x,y*)has a critical point at (–1/3,1/6). Use the second derivative test to demonstrate that for*k*= –1/2*, f*has local maximum at (–1/3,1/6). Use algebra and/or calculus techniques to justify all work. - Acknowledge sources, using APA-formatted in-text citations and references, for content that is quoted, paraphrased, or summarized.
- Demonstrate professional communication in the content and presentation of your submission.