Calculus Answers

If B(u)

 is a differentiable vector function of u

 and ||B(u)||=1

, prove that du

 is perpendicular to B

Let E:=E

x

i

^

+E

y

j

^

+E

z

k

^

 and H:=H

x

i

^

+H

y

j

^

+H

z

k

^

 be two vectors assumed to have continuous partial derivatives (of second order at least) with

respect to position and time. Suppose further that E

 and H

 satisfy the equations:

∇⋅E=0,∇⋅H=0,∇×E=−1

c

∂H

∂t

,∇×H=1

c

∂E

∂t

prove that E

 and H

 satisfy the equation

∇

2

E

i

=1

c

2

∂

2

E

i

∂t

2

 and ∇

2

H

i

=1

c

2

∂

2

H

i

∂t

2

Here, i=x,y

 or z

.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇

2

V.

Evaluate: ∫4 𝑥𝑒^𝑥 dx.

Calculate the area under the curve 𝑦 = 𝑥³ + 4𝑥 + 1 from x=-3 to x=3

1.    Evaluate :        (𝑥³ + 1

𝑥3

) dx.

Evaluate maximum and minimum value of the function

ƒ(𝑥) =  𝑥³-3𝑥² + 3𝑥+1

1.    Also find the equation

of tangent and normal of the ellipse

𝑥2

4

𝑦2

+

16

=1 at the point (-1,3).

b) Evaluate the 2^nd order partial derivatives 62 u

6𝑥2

62 u

and 6𝑦2 if

𝑢 = 2𝑥³+3𝑥² 𝑦 + 𝑥𝑦³+𝑦³.

1.    Find the derivatives of the following functions with respect to x.

𝑥³ + 𝑦³ = 3   

𝑦 = (sin 𝑥)^{𝑡𝑎𝑛𝑥}

A spring is such that a 2 lb weight stretches it by 6 in. An impressed force of F(t) = ¼ sin 8t is

acting on the spring. If the 2-lb weight is released from a point 3 in below the equilibrium point,

describe the motion.