Given the magnitude of vector u is 12 and theta is 20 degrees, find the x and y component of vector u.
In order to solve this problem we first need to define our equations to solve for the x and y components of vector u. x= (magnitude of vector u)*cos(theta) y= (magnitude of vector u)*sin(theta) Since we know the magnitude of vector u and theta we just need to fill in the values to our equation x= 12*cos(20)= 11.28 y= 12*sin(20)= 4.10 ***Since our theta is in degrees make sure that your calculator is set to DEG not RAD!!
If f(x)= (4x^2+7)^3 what is f'(x)?
In order to find the derivative of f(x) we first need to evaluate what rule we are going to use. The key gives away here is that there is a function inside of a function. Therefore, we are going to use the chain rule. Our chain rule states that: f'(x)= h'(g(x))*g'(x) Where our h(x) represents the outside function (or the function that is holding the inner function) and our g(x) represents the inside function. Let's go back to our function f(x)=(4x^2+7)^3. We need to figure out which part is our h(x) and which part is our g(x). If I were to replace everything on the inside of the parenthesis for f(x) with just the variable x then we would simply have x^3. This is what we are going to use for h(x) because it is representing what is holding the inside function. Therefore we have: h(x)= x^3 Now, everything on the inside of the parenthesis for f(x) is (4x^2 +7). This is what we are going to use for g(x) because it's the inner function. Therefore we have: g(x)= (4x^2 +7) Now that we have defined our h(x) and our g(x) we need to find our h'(x) and our g'(x). Once we find these expressions we then simply plug them back into our original chain rule equation. h'(x) and g'(x) can now be solved by using the power rule. To quickly refresh on the power rule if f(x)= x^n then f'(x)= nx^n-1. h(x)= x^3 g(x)= 4x^2 + 7 h'(x)= 3x^2 g'(x)= 8x **** Do not forget that the derivative of a constant is 0. That is why g'(x) is simply 8x and the 7 goes away**** Now that we have our expressions we can plug them into our chain rule equation: f'(x)= h'(g(x))*g'(x) f'(x)= 3(4x^2 +7)^2*(8x), now all we have to do is simplify f'(x)= 24x(4x^2 +7)
Use the points (4,10) and (2,2) to express and the equation of a line.
First, we should remember the standard form of an equation of a line which is: y-y1=m(x-x1) Second, we need to solve for our slope, m: m= (y1-y2)/(x1-x2) Our first point is (4,10) which is going to represent (x1,y1). Our second point is (2,2) which is going to represent (x2,y2). So now we can solve for our slope m. Recall m= (y1-y2)/(x1-x2) m= (10-2)/(4-2) = (8/2) = 4 Now that we know our slope (m) is 4 we can substitute the values y1, x1, m into our standard form of a line. Remember standard form is y-y1=m(x-x1) y-10 = 4(x-4), we need to distribute the 4 on the right side of the equation y-10 = 4x-16, now we need to simplify our equation by adding the 10 to both sides of the equation This brings us to our final answer: y = 4x - 6