Calculus for physics problems

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Answer the following questions on a separate sheet of paper. Show your work and box your
answers.
1. (10 points) Using the following matrices:
4 1 -2
?= 8 0 0
4 2 -5
 ? =
-3 0
3 6
0 0
1
0
1
a. Calculate AB.
b. Calculate A+B.
c. Calculate the determinant of A.
2. (10 points) Using the vectors,
? = 5? – 3? + 2?  
 ? = -7? – 2? + 3?
a. Calculate the dot product ? · ?.
b. Calculate the cross product ?×?.
3. (10 points) A ray is at -15 mm with slope -0.5.
a. It passes through 400 mm of empty space. Describe the ray after this occurs.
b. The ray from part (a) passes through a lens of focal length -200 mm. Describe the ray
after this occurs.
4. (10 points) Rotate the vector ? = 2? + 3? – ?
– 45° (– p/4) about the x axis.
5. (10 points) Using the following matrix:
3
2
1
4
a. Find the eigenvalues.
b. Find the eigenvectors. (Label which eigenvector corresponds to which eigenvalue.)
6. (5 points) Calculate ?? ?, ?, ? for
? ?, ?, ? = ?? ! – 5? ! sin ?
7. (20 points) For the vector field ? = ?? – ??,
a. Sketch ?.
b. Calculate the divergence.
c. Calculate the curl.
d. Explain why the divergence and curl that you calculated make sense for your sketch.
8. (10 points) For ? = ??? + ??? – 2?? ! ?, calculate the line integral along the curved line path
? = -? ! between the point (1, -1, 2) and the point (2, -4, 2).
9. (10 points) For ? = ??? + ??? – 2?? ! ?, calculate the surface integral for the triangular
surface in the plane ? = 3, bounded by the x axis, the line ? = 1 and the line ? = -2?.
10. (5 points) Calculate the volume integral for the scalar function ? = ? ! ?(4 – ? ! ) over the
cube where x runs from 0 to 1, y runs from 3 to 8, and z runs from 1 to 4.
11. (12.5 points) Find the general solution of the following differential equation:
?! ?
??
+2
+ 17? = 0
!
??
??
12. (12.5 points) Find the general solution of the following differential equation:
?! ?
??
-3
– 10? = 0
!
??
??
13. (12.5 points) Find the general solution of the following differential equation:
?! = –
2?
? + ?! + 3
?! + 1
14. (12.5 points) Find the general solution of the following differential equation:
3??’ – ? = ? + 1

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