Differential Equations
The derivative of y with respect to x:
dy/dx = x + 4
Answer:
y = x2/2 + 4x + C
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..)
Finding the derivative of the numerator and denominator to evaluate the limit of a function.
L'hopital's rule
What is the antiderivative of eu?
eu+C
Derive the following:
y=9x3+5x2+7x+3
y'=27x2+10x+7
Solve the following Differential Equation:
y' = 7x/y
Answer:
y2 - 7x2 = C
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
What is the formal term that accurately portrays a hole in the graph?
Removable Discontinuity
What is the derivative of ln(u)?
(u')/(u)
y=2x
y'=(2x)(ln2)
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.
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)
What is amplitude?
The distance a function is from the axis.
What is the antiderivative of (1/u)du?
ln|u|+C
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.
Solve the following particular solution:
dy/dx = 3y ; y(0) = 2
When t = 4?
Answer:
t = In 2 / 3
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)
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.
What is the derivative of arc cot(u)?
(-u')/(1+u2)
Derive the following:
cos(8x)2
-16(cos(8x))sin(8x)
Solve the following differential equation:
y' - 16xy = 0
Answer:
y = Ce8x^2 or y = e8x^2 + C
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)
What is a branch of mathematics involving derivatives and integrals, used to study motion and changing values?
Calculus.
What is the antiderivative of (cot(u)) du?
ln|sin(u)|+C
(d)/(dx) arc tan(3x)= (3)/(1+9x2)