Calculus Round 2

Differential Equations

100

The derivative of y with respect to x:

dy/dx = x + 4

Answer:

y = x²/2 + 4x + C

200

Solve the following Differential Equation: 

y' = 7x/y 

Answer:

y² - 7x² = C

300

Can this differential equation be solved by using separation of variables? Why?

dy/dx = 2x + 3y + 6 

No. It's not possible to factor 2x + 3y + 6 as f(y)g(x). Thus, it's not possible to bring the equation to the form dy/dx = f(x)g(y). The equation is not seperable. 

400

Solve the following particular solution: 

dy/dx = 3y ; y(0) = 2

 When t = 4?

Answer: 

t = In 2 / 3

500

Solve the following differential equation:

y' - 16xy = 0 

Answer: 

y = Ce^{8x^2}   or  y = e^{8x^2 + C}

Applications of Integration

100

Find the volume of the solid generated by revolving the region bounded by the graph of the equation about the x-axis. 

y = 1/x, y=0, x=1, x=3 

Answer: 

V = 2/3(3.141..)

200

Find the volume of the solid generated by revolving the region bounded by the graph of the equation about the y-axis.

y = 3(2 - x), y=0, x=0 

Answer: 

V = 1/3(3.141)r²h

300

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. 

y= x², y = 4x - x², line x = 4   

Answer: 

V = 16(3.141)

400

Find the area bounded by the curves (in the 1st quadrant): 

y=cos((3.141)x/2) and y=1 - x² 

Answer: 

A= 2/3 - 2/(3.141)

500

 An object moves along a straight line with acceleration given by a(t) = 1-sin((3.141)t). Assume that when t = 0, s(t) = v(t) = 0.  Find s(t) and v(t). 

Answer: 

s(t) = t²/ 2 + sin((3.141)t)/(3.141)² - t/(3.141)

v(t) = t + cos((3.141)t)/(3.141) - 1/(3.141)

Vocabulary

100

Finding the derivative of the numerator and denominator to evaluate the limit of a function.

L'hopital's rule

200

What is the formal term that accurately portrays a hole in the graph? 

Removable Discontinuity

300

What is amplitude? 

The distance a function is from the axis.

400

What is the proper term that describes an object with respect to an initial point that is defined as the vector distance from the initial point to the final point?

Displacement implies that an object has been moved, or has been displaced. Displacement is defined to be the change in position of an object.

500

What is a branch of mathematics involving derivatives and integrals, used to study motion and changing values? 

Calculus. 

Formulas

100

What is the antiderivative of e^u?

e^u+C

200

What is the derivative of ln(u)?

(u')/(u)

300

What is the antiderivative of (1/u)du?

ln|u|+C

400

What is the derivative of arc cot(u)?

(-u')/(1+u²)

500

What is the antiderivative of (cot(u)) du?

ln|sin(u)|+C

Miscellaneous

100

Derive the following:

y=9x³+5x²+7x+3

y'=27x²+10x+7

200

y=2^x

y'=(2^x)(ln2)

300

List the Degree Rules

If the degree of the top is less than the degree of the bottom, the limit will equal zero.

If the degree of the top is equal to the degree of the bottom, the limit will be the ratio of the leading coefficients.

If the degree of the top is greater than the degree of the bottom, the limit does not exist.

400

Derive the following:

cos(8x)²

-16(cos(8x))sin(8x)

500

Derive the following:

(d)/(dx) arc tan(3x)

(d)/(dx) arc tan(3x)= (3)/(1+9x²)