I need help with calculus summer homework!

Question

1. Describe the components of the vector v(t)=<t²,5t> and the meaning of this vector.

 

2. Convert the following parametric equation into a rectangular equation:

{x(t)=5cost}      

{y(t)=-5sint}      (these are meant to be in the same bracket, not two separate}

 

3. A plane is flying along a bearing of 60 degrees at 500 mph while the wind is blowing at a bearing of 315 degrees at 40 mph. 

a. draw a diagram modeling the situation

b. what is the speed and direction of the vector on which the plane is flying?

 

4. Given the following terms, find the general term for the series:

1+ (-1x)/2 + (x²)/8 + (-x³)/48 + (x⁴)/384

Answer 1

v(t)=(t²,5t)

 

Part 1

 

x component is a parabola with vertex at (0,0) and opening upward.

 

y component is a line from (0,0) with a slope of 5.

 

Part 2

 

x(t)=5cos(t)

y(t)=-5sin(t)

 

x²+y²=25[cos²(t)+sin²(t)]=25(1)=25

 

This is the equation of a circle with radius 5 and center at (0,0)

 

Part 3

 

Bearings are measured relative to North with positive angles in the clockwise direction.

 

cos(60)=1/2
sin(60)=(√3)/2

cos(315)=1/√2
sin(315)=-1/√2

 

A vector with length 500 and angle 60 relative to north has x,y coordinates of (500sin(60),500cos(60))

 

A vector with length 40 and angle 315 relative to north has x,y coordinates of (40sin(315),40cos(315))

 

(500sin(60),500cos(60))=(500(√3)/2,500/2)=(250√3,250)

 

(40sin(315),40cos(315))=(-40√2,40/√2)

 

Add the vectors to get a resultant vector with x,y coordinates of

 

(250√3-40√2,250+40/√2)=(221.716,278.284)

 

The speed is the magnitude of the resultant vector.

 

Speed=sqrt(278.284²+221.716²)=355.809

 

The direction is obtained by observing that the tangent of the angle is 278.284/221.716=1.225

 

The angle is arctan(1.225)=50.77 degrees (a northeast direction)

 

A general term that fits the terms that are given is

 

a₀=(-1)⁰x⁰/[2⁰(0!)=1

a₁=(-1)¹x¹/[2¹(1!)=-x/2

a₂=(-1)²x²/[2²(2!)=x²/8

 

a_i=(-1)ⁱxⁱ/[2ⁱ(i!)

 

Double check the above to make sure that there are no errors.

 

cos(315)=1/√2

sin(315)=-1/√2