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Homework Set 1 for MATH 2850 Due: September 4, 2018 Textbook: Thomas’s Calculus, 12th Edition
In Exercises 1 , 3 , 10 , and 12 , match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsopid, etc.) The surfaces are labeled (a)-(f).
- x^2 + y^2 + 4z^2 = 10
- 9 y^2 + z^2 = 16
- z = − 4 x^2 − y^2
- 9 x^2 + 4y^2 + 2z^2 = 36
In Exercises 2 and 4 , r(t) is the position of a particle in the xy-plane at time t. (1) Find an equation in x and y whose graph is the path of the particle. (2) Find the particle’s velocity and acceleration vectors at the given value of t.
- r(t) =
t t + 1
i +
t
j, t = − 1 / 2
- r(t) = (cos 2t) i + (3 sin 2t) j, t = 0
In Exercises 11 and 13 , r(t) is the position of a particle in space at time t. (1) Find the particle’s velocity and acceleration vectors. (2) Find the particle’s speed and direction of motion at the given value of t.
- r(t) = (2 cos t)i + (3 sin t)j + 4tk, t = π/ 2
- r(t) = (2 ln (t + 1)) i + t^2 j +
t^2 2
k, t = 1
In Exerciese 20 and 21 , find parametric equations for the line that is tangent to the given curve at the given parameter value t = t 0.
- r(t) = t^2 i + (2t − 1)j + t^3 k, t 0 = 2
- r(t) = ln t i +
t − 1 t + 2
j + t ln t k, t 0 = 1
Evaluate the integrals in 2 , 4 , 5 , 8 , 9 :
∫ (^2)
1
[ (6 − 6 t) i + 3
t j +
( 4 t^2
) k
] dt
∫ (^) π/ 3
0
[(sec t tan t) i + (tan t) j + (2 sin t cos t) k] dt
∫ (^4)
1
[ 1 t
i +
5 − t
j +
2 t
k
] dt
∫ (^) ln 3
1
[tet^ i + et^ j + ln t k] dt
∫ (^) π/ 2
0
[ (cos t) i − (sin 2t) j + (sin^2 t) k
] dt
