preTEST1A Solved Multivariable Calculus
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The document contains a 24-page math exam with 6 questions testing skills in limits of rational functions, difference quotients, vector arithmetic, matrix operations for solving circuit analysis problems, properties of tetrahedrons, derivatives and integrals of position/velocity/acceleration functions, area of triangles in space, and vector calculus identities. It provides a reference sheet of important derivative formulas and defines the difference quotient used to numerically estimate derivatives.
preTEST1A Solved Multivariable Calculus
- 1. MAT225 TEST1A Name: Show all work algebraically if possible. RVA/RVB (Question 1) Rational Functions Let f(x)= x − 4 2 2x − 16 3 (1a) (x) ? lim x→−∞ f = (1b) (x) ? lim x→∞ f = (1c) (x) ? lim x→−2− f = (1d) (x) ? lim x→−2+ f = (1e) (x) ? lim x→2− f = (1f) (x) ? lim x→2+ f = TEST1A page: 1
- 2. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 2
- 3. MAT225 TEST1A Name: Show all work algebraically if possible. RVA/RVB (Question 2) Difference Quotients Let g(x) = x2 - x (2a) Find g’(x) using the Difference Quotient. (2b) Calculate g(x) and g’(x) when x= . 2 1 (2c) State the equation of the tangent line to g(x) at x= . 2 1 TEST1A page: 3
- 4. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 4
- 5. MAT225 TEST1A Name: Show all work algebraically if possible. RVA/RVB (Question 3) Difference Quotients (1a) = ? lim h→0 h (x+h) − x 2 2 (1b) = ? lim h→0 h (5+h) − 25 2 (1c) = ? lim h→0 h sin(x+h) − sin(x) (1d) = ? lim h→0 h sin( +h) − 1 2 π (1e) = ? lim h→0 h e − e x+h x (1f) = ? lim h→0 h e − 1 h TEST1A page: 5
- 6. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 6
- 7. MAT225 TEST1A Name: Show all work algebraically if possible. (1) Vector Arithmetic Let u = <1,2,3>, v = <3,–2,1>, w = <4,0,6> (1a) Find u – v (1b) Find –2(u – v) (1c) Find u + w (1d) Find 3(u + w) (1e) Find 3(u + v) – 2(u – v) TEST1A page: 7
- 8. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 8
- 9. MAT225 TEST1A Name: Show all work algebraically if possible. (2) Circuit Analysis: The currents I1 , I2 , I3 in a circuit with resistors R1, R2 and voltages E1 , E2 are described by the following system of equations: R1I1 + R3I3 = E1 R2I2 + R3I3 = E2 I1 + I2 - I3 = 0 Let R1 = 2 Ohms, R2 = 1 Ohm, R3 = 4 Ohms, E1 = 14 Volts and E2 = 28 Volts. A = X = B = (2a) If AX = B, list the Minors of Matrix A. (2b) If AX = B, state the CoFactors of the Minors of Matrix A. (2c) If AX = B, find the Transpose of the CoFactors of the Minors of Matrix A. TEST1A page: 9
- 10. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 10
- 11. MAT225 TEST1A Name: Show all work algebraically if possible. (2) Circuit Analysis: The currents I1 , I2 , I3 in a circuit with resistors R1, R2 and voltages E1 , E2 are described by the following system of equations: R1I1 + R3I3 = E1 R2I2 + R3I3 = E2 I1 + I2 - I3 = 0 Let R1 = 2 Ohms, R2 = 1 Ohm, R3 = 4 Ohms, E1 = 14 Volts and E2 = 28 Volts. A = X = B = (2d) If AX = B, calculate det(A) = |A|. (2e) If AX = B, find A-1 . TEST1A page: 11
- 12. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 12
- 13. MAT225 TEST1A Name: Show all work algebraically if possible. (2) Circuit Analysis: The currents I1 , I2 , I3 in a circuit with resistors R1, R2 and voltages E1 , E2 are described by the following system of equations: R1I1 + R3I3 = E1 R2I2 + R3I3 = E2 I1 + I2 - I3 = 0 Let R1 = 2 Ohms, R2 = 1 Ohm, R3 = 4 Ohms, E1 = 14 Volts and E2 = 28 Volts. A = X = B = (2f) Rewrite AX = B as X=A-1 B. (2g) If AX = B, find the solution vector X. TEST1A page: 13
- 14. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 14
- 15. MAT225 TEST1A Name: Show all work algebraically if possible. (3) TetraHedrons A molecule of Methane CH4 forms a Tetrahedron with Carbon at the centroid E(k/2, k/2, k/2) and Hydrogen atoms at the 4 corners: A(0, 0, 0), B(k, k, 0), C(k,0,k) and D(0, k, k) (3a) Find the length of side CD. (3b) What is the angle between sides AC and AD. (3c) Calculate the bonding angle between the vectors ED and EA. TEST1A page: 15
- 16. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 16
- 17. MAT225 TEST1A Name: Show all work algebraically if possible. (4) r(t) , v(t) , a(t) The motion of a particle is given by the position vector r(t) = < 3cos(t), 3sin(t), t> (4a) Find v(t) and its magnitude and interpret your result. (4b) Find a(t) and its magnitude and interpret your result. (4c) Calculate the dot product v(t) • a(t) and interpret your result. TEST1A page: 17
- 18. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 18
- 19. MAT225 TEST1A Name: Show all work algebraically if possible. (5) Triangles In Space Consider the triangle with vertices A(2, 1, 0), B(1, 0, 1) and C(2, -1, 1) (5a) Find the area of the triangle ABC. (5b) What is the equation of the plane containing points A, B and C? (5c) Where is the point of intersection of this plane with the line parallel to the vector v = <1, 1, 1> passing through the point S(-1, 0, 0)? TEST1A page: 19
- 20. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 20
- 21. MAT225 TEST1A Name: Show all work algebraically if possible. (6) Vector Calculus Let r(t) be a displacement vector in space. (6a) Find the derivative of r(t) • r(t) with respect to time. (6b) Show that r(t) and v(t) are perpendicular when r(t) is constant. (6c) Calculate r(t) • a(t) given r(t) is constant. TEST1A page: 21
- 22. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 22
- 23. MAT225 TEST1A Name: Show all work algebraically if possible. Reference Sheet: Derivatives You Should Know Cold! Power Functions: x nx d dx n = n−1 Trig Functions: sin(x) os(x) d dx = c cos(x) in(x) d dx = − s tan(x) (x) d dx = sec2 cot(x) (x) d dx = − csc2 sec(x) ec(x) tan(x) d dx = s csc(x) sc(x) cot(x) d dx = − c Transcendental Functions: e d dx x = ex a n(a) a d dx x = l x ln(x) d dx = x 1 log (x) d dx a = 1 ln(a) x 1 Inverse Trig Functions: sin (x) d dx −1 = 1 √1−x2 cos (x) d dx −1 = −1 √1−x2 tan (x) d dx −1 = 1 1+x2 cot (x) d dx −1 = −1 1+x2 Product Rule: f(x) g(x) (x) g (x) (x) f (x) d dx = f ′ + g ′ Quotient Rule: d dx f(x) g(x) = g (x) 2 g(x) f (x) − f(x) g (x) ′ ′ Chain Rule: f(g(x)) (g(x)) g (x) d dx = f′ ′ Difference Quotient: f’(x) = lim h→0 h f(x+h) − f(x) TEST1A page: 23
- 24. MAT225 TEST1A Name: Show all work algebraically if possible. TEST1A page: 24