preTEST3A Double Integrals Solved
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This document contains a test with 8 multi-part math problems involving concepts like derivatives, integrals, vectors, and geometry. The test asks the student to show their work algebraically and provides reference sheets on important derivatives and integrals. It provides context for the high-level concepts being assessed but does not give the answers to the problems.
![MAT225 TEST3A Name:
Show all work algebraically if possible.
RVF (Question 1) s(t), v(t), a(t)
(1) A bullet is shot upward from the surface of the Moon such that
(t) 60t .8t
y = 1 − 0 2
[y] = meters, [t] = seconds, t≥0.
(1a) Find y‘(t)
(1b) Calculate y‘(0)
(1c) Solve for t when y’(t) = 0.
(1d) What is the maximum height?
(1e) How fast is the bullet moving when it hits the ground?
TEST3A page: 1](https://image.slidesharecdn.com/2020pretest3asolved-errata-210326134644/85/preTEST3A-Double-Integrals-Solved-1-320.jpg)
preTEST3A Double Integrals Solved
- 1. MAT225 TEST3A Name: Show all work algebraically if possible. RVF (Question 1) s(t), v(t), a(t) (1) A bullet is shot upward from the surface of the Moon such that (t) 60t .8t y = 1 − 0 2 [y] = meters, [t] = seconds, t≥0. (1a) Find y‘(t) (1b) Calculate y‘(0) (1c) Solve for t when y’(t) = 0. (1d) What is the maximum height? (1e) How fast is the bullet moving when it hits the ground? TEST3A page: 1
- 2. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 2
- 3. MAT225 TEST3A Name: Show all work algebraically if possible. RVG (Question 2) a(t), v(t), s(t) (2) Find f(x) such that f(x) is a function defined for all with these properties: − x > 5 (i) f ”(x) = 1 3√x+5 (ii) tangent line to the graph of f at (4,2) makes a 45° angle with the x-axis. TEST3A page: 3
- 4. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 4
- 5. MAT225 TEST3A Name: Show all work algebraically if possible. (1) Dot Products Given the triangle ABC, A(1,1), B(4,1) and C(4,4): (1a) Find the components of the vectors AB and AC. (1b) Calculate the magnitudes of the vectors AB and AC. (1c) Use the Dot Product of AB and AC to find the measure of angle A. (1d) What is the area of ABC? Δ (1e) Let , does equal the triangle area? s = 2 a+b+c √s(s )(s )(s ) − a − b − c TEST3A page: 5
- 6. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 6
- 7. MAT225 TEST3A Name: Show all work algebraically if possible. (2) Cross Products Given the triangle ABC, A(1,1), B(4,1) and C(4,4): (2a) Find the following Cross Products: OB x OC, OC x OA, OA x OB. (2b) Sum the following Cross Products: OB x OC, OC x OA, OA x OB. (2c) What is the magnitude of the sum of these Cross Products? (2d) Is there a relationship between this magnitude and the triangle area? TEST3A page: 7
- 8. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 8
- 9. MAT225 TEST3A Name: Show all work algebraically if possible. (3) Determinants Let the vectors u = <1,2,3>, v=<1,0,1> and w=<2,3,4>: (3a) Find u • (v x w). (3b) What does this value measure? TEST3A page: 9
- 10. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 10
- 11. MAT225 TEST3A Name: Show all work algebraically if possible. (4) Determinants Let the vectors u = <1,2,3>, v=<1,0,1> and w=<2,3,4>: (4a) Find det(u,v,w). (4b) What does this value measure? TEST3A page: 11
- 12. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 12
- 13. MAT225 TEST3A Name: Show all work algebraically if possible. (5) Iterated Integrals Consider the area in the xy-plane bounded by: , − √4 − y2 ≤ x ≤ √4 − y2 . 0 ≤ y ≤ 2 ydx A = ∫ 2 −2 ∫ √4−y2 0 d (5a) Draw this region labeling a vertical Riemann Rectangle with thickness dx. (5b) Explain how to setup this integral in terms of dydx to calculate the area. (5c) Evaluate your integral in terms of dydx. TEST3A page: 13
- 14. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 14
- 15. MAT225 TEST3A Name: Show all work algebraically if possible. (5) Iterated Integrals Consider the area in the xy-plane bounded by: , − √4 − y2 ≤ x ≤ √4 − y2 . 0 ≤ y ≤ 2 ydx A = ∫ 2 −2 ∫ √4−y2 0 d (5d) ReWrite this integral in terms of dxdy to calculate the area. (5e) ReEvaluate your integral in terms of dxdy. (5f) ReWrite this integral in terms of to calculate th area. drdθ r (5g) ReEvaluate your integral in terms of drdθ. r TEST3A page: 15
- 16. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 16
- 17. MAT225 TEST3A Name: Show all work algebraically if possible. (6) Iterated Integrals Find the volume of the solid bounded by the surface f(x,y)=1-xy above the triangle bounded by y=x, y=1 and x=0. (1 y) dxdy V = ∫ 1 0 ∫ y 0 − x (6a) Explain how to set up an integral to calculate this volume in terms of dxdy. (6b) Evaluate your double integral. (6c) ReWrite the integral terms of dydx. (6d) Evaluate your double integral. TEST3A page: 17
- 18. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 18
- 19. MAT225 TEST3A Name: Show all work algebraically if possible. (7) Line Integrals Calculate a line integral to find the mass of a wire given density and the path C: ρ Density Function (x, ) y ρ = F y = x Along the path C: r(t)=<4t, 3t> such that 0 ≤ t ≤ 1 (7a) Find ds=|r’(t)|dt (7b) Write the line integral in terms of t. ds ∫ C F (7c) Evaluate your integral. TEST3A page: 19
- 20. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 20
- 21. MAT225 TEST3A Name: Show all work algebraically if possible. (8) Work Done By A Conservative Field Given the Vector Field (x, ) < x y , 2x xy F y = a 2 + y3 + 1 3 + b 2 + 2 > (8a) Find the values of and b for which F is conservative. (8b) Using these values of a and b, find f(x,y) such that F = gradient(f). (8c) Find the work done through F along the curve C: (t) cos(t), y(t) sin(t), 0 x = et = et ≤ t ≤ π TEST3A page: 21
- 22. MAT225 TEST3A Name: Show all work algebraically if possible. TEST3A page: 22
- 23. MAT225 TEST3A 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) TEST3A page: 23
- 24. MAT225 TEST3A Name: Show all work algebraically if possible. Reference Sheet: Anti-Derivatives You Should Know Cold! Power Functions: dx x ∫xn = n n−1 Trig Functions: os(x)dx in(x) ∫c = s + C in(x)dx os(x) ∫s = − c + C ec (x)dx an(x) ∫s 2 = t + C sc (x)dx ot(x) ∫c 2 = − c + C ec(x)tan(x)dx ec(x) ∫s = s + C sc(x)cos(x)dx sc(x) ∫c = − c + C Transcendental Functions: dx e ∫ex = x + C dx ∫ax = ax ln(a) + C dx n(x) ∫ x 1 = l + C dx log (x) ∫ 1 ln(a) x 1 = a + C Inverse Trig Functions: dx sin (x) ∫ 1 √1−x2 = −1 + C dx cos(x) ∫ −1 √1−x2 = + C dx tan (x) ∫ 1 1+x2 = −1 + C dx cot (x) ∫ −1 1+x2 = −1 + C Integration By Parts (Product Rule): dv uv du ∫u = −∫v + C Integration By Partial Fractions Example (Quotient Rule): ∫ dx x(x+1) = ∫ x Adx +∫ x+1 Bdx TEST3A page: 24