That's a great start and you have almost solved the problem.
You found the critical point at (0,1), but you need to find the discriminant in order to determine if this is the max or min of the function. The discriminant is found as fxxfyy -fxy2 = 6*4 = 24. Since...
The rational zeros theorem is a method to determine the possible roots of a function. The rational zeros are p/q, where p are all of the factors of the constant term, and q are all of the factors of the leading coefficient.
For this function, p = +/- 28, +/- 14, +/- 7,...
When the boat is traveling upstream, it's speed is v - 3, and similarly for downstream its v + 3, where v is the speed of the boat relative to the water.
The time to go upstream is 3/(v-3) and to go downstream its 3/(v+3). With total time = 8, we have
3/(v-3) + 3/(v+3)...
For a function f(x,y,z) = 0, the directional derivative is given by the gradient of the function, <fx,fy,fz>. Since a function defined as z = f(x,y), this is essentially equivalent to 0 = f(x,y) - z, then the gradient is <fx,fy,-1>.
b) The gradient will be normal...
The general form for an exponential growth equation is P(t) = Ae^(rt), where P(t) is the number of bacteria after time t, A is the initial number, and r is the growth rate.
If they quadruple every hour, then P/A = 4 when t = 1.
So 4 = e^(r*1). Taking LN of both sides gives
Any time you have want to factor an expression in the form (a - b), you can always break it into
(sqrt(a) + sqrt(b)) (sqrt(a) - sqrt(b))
In this case, "a" would be 100x^2 and "b" would be 1.
So it can be factored to (10x + 1) (10x - 1...
To factor this completely, you start by determining the first, most apparent factor, which you get by setting the expression equal to zero and solving for z.
You get z^3 = -125 ------>> z = -5
Therefore, one of your factors is (z+5).
To get the total net worth, simply multiply the average net worth by the number of families. Using scientific notation,
410,000 = 4.1*10^5
(4.1*10^5)*(6.1*10^7) = 4.1*6.1 * 10^12 = 25.01 * 10^12 = 2.5 * 10^13 (Rounded to the nearest tenth)
First of all, you need to determine if this parabola is opening vertically or horizontally. Since the vertex and the focus share the same x-value, the line of symmetry is at x = 3, which is vertical. The standard form of a vertical parabola is.
4p*(y-k) = (x - h)^2, where (h,k) are...
sqrt(4) = 2.
For 2 - x/(x-2) = -2/(x-2) start by multiplying through by (x-2) to get the variable out of the denominator. This gives:
2*(x-2) - x = -2. Expand this out, then combine the constant terms on one side and variable terms on the other.
2*x - 2*2 - x = -2
Don't be intimidated by numbers with decimals, they are still just constants and treated like any other.
So to solve, you want to isolate the variable x.
(1249/x)/6.849 = 39.08 > Multiply both sides by 6.849
1249/x = 39.08*6.849 > Flip both sides...
The reason you subtract 23 from both sides is to isolate the variable, x. By having x by itself, you have answered the question, "what value of x will make this equation true?"
Beyond that, the only reason for doing it is to learn and understand the underlying logic of...
Since both equations have 3a on the left side, we can simply set the expressions on the right side equal to eachother. i.e.
3a = 5-b = 4 -b
Adding b to both sides and subtracting 4, we get
5 - b + b - 4 = 4 - b + b - 4
1 = 0
Since this is false, there are...
It's generally going to be easiest to complete the square if the coefficient in front of the x^2 (or in this case, q^2) term is 1, so start by multiplying through by 2. This gives:
q^2 - 5/2q -6 = 0.
Then, take the coefficient of the linear term (-5/2), divide it by 2,...
-10 + (-5) + 5...+n
= -10 + (-10+5*1) + (-10+5*2) + (-10 + 5*3)....(-10+5*(n-1))
= -10*n + 5*(n-1)*n/2
For n = 19, =-10*19 + 5*18*19/2 = 665.
For Sn = 375, solve for n to get n = 15.
Start by finding the parametric form of the curves that bound the surface. The 3 corners of the tetrahedron occur with the intersection of each coordinate axis, so simply set 2 of the variables equal to zero to get the intercept.
P1 = (2,0,0) P2 = (0,2,0) P3...
Assuming you meant f(x) = x^2 + 1.
f + g = x^2 + 1 + (3x - 4) = x^2 + 3x - 3.
f*g(5) = ( 5^2+1)*(3*5-4) = 26*11 = 286.
A = B + 473.
A + B = 119,973.
Substituting the first into the 2nd gives: 2B + 473 = 119973. 2B = 119,500.
B = 59,750.
A = 60,223.
Well, the volume of a sphere is given by 4/3*pi*r^3. Setting r = 7, we get 4/3*pi*7^3 = 1436.75 ~= 1437 cm^3.
When adding numbers with a decimal, be sure to line up the decimal point when you write it out vertically and proceed as usual.