Mathletes Competition

Calculus I

100

Find the derivative of f(x)=x¹⁰

10x⁹

200

Evaluate the limit as x approaches 2 for the function 

8-3x+12x²

50

300

Integrate 

(1/x²)dx

(-1/x)+C

400

What rule is this:
dxd[f(g(x))]=f′(g(x))g′(x)

The chain Rule

500

(d/dx) arcsin(x)= ?

(d/dx) arccos(x)= ?

(d/dx) arctan(x)= ?

{1/ sqrt(1-x²)}

-{1/ sqrt(1-x²)}

1/ (1+x²)

Calculus II

100

Write a formula for the nth term of the given sequence. 

1,2,4,8,16,32

a_n= 2^n-1

200

Find the derivative of each function:

f(x)=6csch³5x

6[3csch²5x *(-csch5xcoth5x)5]

300

Evaluate the integral involving inverse trigonometric functions

"S" means integral

S {(4x)/ sqrt[81-x⁴]} dx

2sin^-1(x²/9) +C

400

Integrate= "S" 

S (sin⁷x/ cos⁴x)dx

(1/3cos³x)-(3/cosx)-3cosx+(1/3)cos³x+C

500

What is the volume of the solid made by rotating y= cubed root x and y= (x/4) in the first quadrant about the y- axis?

V= (512pi/21)

Calculus III

100

Let f(x,y)= x²+xy³

compute f(-3,-2)

33

200

What is the magnitude of the gradient at (0,2)

f(x,y)=3x+y²

5

300

fxxyzz for f(x,y,z)= z³y²ln|x|

(-12zy)/x²

400

Find the angle between vectors

u= <-2,3>        and v= <-4,-1>

70 (1/4)

500

"S" means integral

ss2xy dA in a quart circle centered at the origin and with r= 5

625

Differential Equations

100

What is the name of this differential equation?

x²y"+xy'+4y=0

Cauchy- Euler Equation

200

Find the Wronskian of the functions y₁=e^2x and y₂=e^-4x

-6e^-2x

300

Consider the Differential Equation:

(x³-y³)dx+xy²dy=0

Find the solution to the above DE using the substitutions y=ux and dy=udx+xdu

ln|x|= (-1/3)(y/x)³+C

400

Solve the initial value problem:

y"-6y'+9y=0,             y(0)=1,             y'(0)=8

y=e^3x+5xe^3x

500

Using an integration factor, find the explicit solution to the linear differential equation:

x(dy/dx)+5y= 2

Find the Singular points and any transient terms

y= (2/5)+(C/x⁵)

Singular points: x=0

Transient terms: (C/x⁵)

Bonus

100

Determine whether the sequence converges or diverges 

a_n= 4/(n⁴)

The series converges 

200

What are the two forms for Green's Theorem?

Circulation Curl & Flux Divergence

300

Write out the 7th line of Pascals Triangle

(a+b)⁶=a⁶+6a⁵b+15a⁴b²+20a³b³+15a²b⁴+6ab⁵+b⁶

400

Assume that y₁=e^2x and y₂=e^-4x are two solutions to an ordinary DE, and the Wronskian values of the ODE are W=-6e^-2x, W₁=e^-5x-2e^-6x, and W₂=2-e^x. Use variation of parameters to find the solution of this DE.

y= C₁e^2x+C₂e^-4x+(1/9)e^-x-(1/4)e^-2x

500

Give the equation for the Elliptic Paraboloid.

z=

z= (x²/a²) + (y²/b²)