Calculus Answers

6 If A=xz^3i−2x^2yzj+2yz^4, find ▽×A at point (1,-1,1).
(a) 2j+3k
(b) 2i+j74k
(c) i+3j+5k
(d) 3j+4k

7 Given that A=A1i+A2j+A3k and r=xi+yj+zk , evaluate ▽⋅(A×r) if ▽×A=0
(a) 0
(b) 3
(c) 2
(d) 5

8 Let A=x^2yi−2xzj+2yzk, find Curl curl A.
(a) 3j+4k
(b) 2x+2)k
(c) (2x+2)j
(d) 3j−4k

9 Given A=2x^2i−3yzj+xz^2k and ϕ=2z−x^3y, find A⋅▽ϕ at point (1,-1,1).
(a) 5
(b) 3
(c) 4
(d) 1

10 Find the directional derivative of ϕ=x^2yz+4xz^2 at (1,-2,-1) in the direction 2i−j−2k
(a) 37/3
(b) 35/3
(c) 25/3
(d) 11/3

1 If ϕ=2xz4−x2y, find |▽ϕ|
(a) √(93)
(b) √(80)
(c) √(12)
(d) √(110)

2 If ϕ(x,y,z)=3x^2y−y^3z^2, find ▽ϕ at point (1,-2,-1)
(a) −12i−9j−16k
(b) i−3j−k
(c) 2i−5j−6k
(d) −3i−4j−2k

3 Find a unit normal to the surface x^2y+2xz=4 at point (2,-2,3)
(a) 2/3i−2/3j−2/3k
(b) −1/5i+2/5j+2/5k
(c) −1/3i+2/3j+2/3k
(d) −1/7i+2/7j+2/7k

4 Let ϕ(x,y,z)=xy^2z and A=xzi−xy^2j+yz^2k,find ∂^3/∂x^2∂z(ϕA)
(a) 2i+2j−5k
(b) 5i−k
(c) 4i−2j
(c) i+j

5 Given that ϕ=2x^2y−xz^3 find ▽^2ϕ
(a) 2y−6xz
(b) 4y−6xz
(c) 2y−xz
(d) y+6xz

5. If A=5t^2+tj−t^3k and B=sinti−costj. evaluate d/dt(A⋅B)
(a) (5t^2−1)cost+11tsint
(b) (5t−1)sint+11tcost
(c) −1cost+2tsint
(d) (5t^2−1)sint+11tcost
6 If A=5t^2+tj−t^3k and B=sinti−costj . evaluate d/dt(A×B)
(a) (t^3sint−3t^2cost)i−(t^3cost−3t^2sint)j+(5t^2sint−11tcost−sint)k
(b) (t^2sint−3tcost)i−(t^3cost−3tsint)j+(5sint−11tcost−sint)k
(c) (tsint−3t^2cost)i−(t^3cost−3t^2sint)j+(5t^2cost−11tcost−cost)k
(d) (tcost−3t^2cost)i−(t^3sint−3t^2cost)j+(5t^2cost−11tsint−cost)k
9. Let A=x^2yzi−2xz^3j−xz^2 and B=4zi+yj+4x^2k , find ∂^2/∂x∂y(A×B) at (1,0,-2)
(a) 2i−8j
(b) −4i−8j
(c) −i−3j
(d) 5i−2j

1. Given that A=sinti + costj + tk , evaluate ∣d2A/dt^2∣
(a) 4 (b)1 (c) 3 (d) 2

3 A particle moves along the curve X = 2t^2, Y = t^2 − 4t and z = 3t − 5, where t is the time. Find the components of the velocity at t = 1 in the direction i −3j+2k

(a) 8√(14)/7

(b) −2√(14)/7

(c) 3√(14)/7

(d) −5√(14)/7

Determine the area under the curve s = 2cos4θ in the range θ = o to π/4 radians

The Discharging characteristic for the capacitive circuit is given by the formula:

v = Ve^-(t/T), where T=CR and is called the time constant

C = 100nF R= 22kOhms and V = 5 V

Differentiate the charging equation and find the rate of change of voltage at t=T

The Voltage, v across the plates of the charging capacitor varies with time, t according to the formula;

v = V (1 - e^-(t/T)), where T = CR and is called the time constant

c = 100nF R = 47kOhms and V = 5V

Differentiate the charging equation and find the rate of change of voltage at 6ms

Integrate by partial fraction (5x + 2) / (3x^2 + x + 4)

find f(x)=7x

Defferentiate the following (2x-7)