So, part A, what we are going to do is, we are given the function of x = 0.13x cubed. And we are trying to find what the derivative of the function is. So, recalling your differential calculus, f of x is 0.13 x cubed. So, f prime of x would be the derivative of 0.13 x cubed. And here, we would take 0.13 outside, because it is a constant. And then we are going to find the derivative of x cubed. And the derivative of x cubed will be 3 times x squared. So, it is 3 times x times whatever is the exponent minus 1 so it’s 2. And that gives us the derivative is 0.39 x squared.

So, in part B, we are asked to find f prime at 7. So, we just found out from the previous problem that f prime of x is 0.39 x squared. So, the derivative of the function at 7 would be 0.39 (7) squared. And that would give us 19.11.

Part C is asking us to find the slope of the tangent line. Slope of tangent line at x = 7. But since we just found out that the slope of the function at 7 is 19.11, that itself is the slope of the tangent line at x = 7.

Part D is asking us to find the angle of the tangent line. And since this is the slope, 19.11, what that means is tangent slope will be equal to 19.11, so that is the angle that the tangent line would be making with the x axis. So if we draw it, I suppose this is our function. This is y. This is x. and this is the function and we want to find out what the value of the tangent line is at x = 7. We just found that to be out. What we want to do is, we want to find the angle the tangent line is making at that particular point. So, we want to find out what this angle is. And this angle, the tangent of that angle is nothing but the slope of the line. So that would be equal to tan inverse of 19.11. And that turns out to be equal to 87.00 degrees. Or if you’re interested in radians it would be 1.519 radians.

Part E is asking us where does the tangent cross the x axis. So, if we look at the tangent lines, it’s a straight line. So, we can say, hey it is given by y=mx+c, the equation of the tangent line. So, if that is the case, then the slope, we have just found out, is 19.11. So, in order to be able to find out what the equation of the tangent line is you need to find c. But what do we know about tangent line? We know one thing about the tangent line: that the tangent line goes through the point 7 comma f of 7 because that’s what we drew the tangent line at. And that’s nothing but the value of the function at 7, which would be 0.13 times 7 cubed because the function is 0.13 x cubed. So that gives us 7 comma 44.59 as the coordinates of one of the points on the tangent line. And that’s the value of the function at 7, would be 44.59. So that allows us to get one more equation here in order to find out c. So, 44.59 is the value of y, m is 19.11 times x, plus c, and x is how much? x is 7. So that is the case we find c equals to minus 81.18. And what that gives us is just the equation of the tangent line. So, the equation of a tangent line is y is equal to 19.11 x minus 89.18. That’s the value m: and the value c. Now we want to find out where does the tangent line cross the x axis. So what would be the value of y at the point where it crosses the x-axis? It’s zero at 19.11 x minus 89.18. That means x is equal to 89.18 over 19.11. And the turns out to be 4.667. This is the point where the tangent line crosses the x-axis. So, if you want to look at it from a graphical point of view: this is our function, which is 0.13 x cubed, and you want to find out where is the tangent line at x equals 7. This is the tangent line. Where does this tangent line cross the x-axis? So, it’s crossing the x-axis at 1.667. And that’s the end of this segment.