If an equation of the tangent line to the curve $ y = f(x) $ at the point where $ a = 2 $ is $ y = 4x - 5 $, find $ f(2) $ and $ f'(2) $.
All right. Uh Let's look at the information that we know. Uh that's given to us. We have a function Y equals F. Of X. Uh That is not the graph of F of X. This blue line is actually degree of of the line that is tangent uh Two degrees off of F of X when X is too. So this line will be tangent uh to the graph of F. Of X when X. S two. Now we don't know what ffx looks like, but here is X equals two. And we know that if we follow from two up until we hit the tangent line. Let's draw a little point in here at this point right here is where this tangent line is going to touch to function. So this line is tangent to the function F of X. When access to. Now, we don't have the graph of F of X. We don't know what it looks like, but we can draw something in just to try to help us out. So let's say let's use uh I use this green color. So, if I want to draw in F a bex. Now, if this line is tangent to the graph of F of X is tangent to the curve of F. Of X. At this point. That means you want the curve to kind of have the same slope. Well, it has to have the same slope at this red point that the that the line does you want to try to draw the function so that it's so that its tangent right there. So let's let's go ahead and draw in the graph of what F of X might look like. All right. So this is what F of X might look like. Now you can see that this blue tangent line, here's the equation of that tangent line over here. But you can see that this blue tangent line is tangent to the curve right there at that red point. So the slope of the green curve right at this red point is the same as the slope of that tangent line. That blue tangent line. Now, we want to answer these two questions. What is the value of our function F When X is too? Well, the function when X is too uh will be the same point when excess to to function value and a change of line have the same value. So to make a long story short uh When X is too You can see that the Y coordinate for this red point is three. So, that means to coordinate, let's pennsylvania coordinates here. The coordinates of this Redpoint, Rx equals two. Why Equal Street? So the point to common three is on the blue tangent line. Uh And it is also on uh green function. So when X is too why is three? And so the function when access to The value of the function is going to be three. Why? Because the point to comic three is on the function. It's on the green curve. So, if a point is on the function, uh then the why coordinate is to function value when X is whatever the X number is. So in this case when X is to y is equal to three. So the function value when x is too is three. In other words F of two is three. So that's why we have F of two equal straight. All right, next. What is f prime of 2? Okay. What is the derivative of our function when excess too? Well, the derivative of your function when X equals two will be the slope of the tangent line to that function when X equals to What is the slope of this blue tangent line? If you remember from algebra Y equals mx plus B. The slope of detainment line is the number multiplying X two, slope is four. So the slope of this blue tangent line is four. So the derivative since this tangent line, since this line is tangent to to function at this point. The derivative of the function when X is to is to slope of the tangent line because the line is tangent to the function right here. When excess to to slope, a detainment line is the derivative of the function at this value of X went access to. So F prime of two is two slope of the tangent line which is for