The Tangent Line Without Calculus One of the goals we have in this course is to use calculus to find the tangent line to a curve at a point. Let's try to do one example without calculus. Let f(x) = x². We haven't completely defined the tangent line yet, but we have seen that "only touches the graph at one point" is not a good enough description. However, in this particular case, it is! Use this graph to convince yourself with geometric intuition that this is true. There is a slider for the value a. If you move this the graph shows you the tangent line at the point (a, a²). Imagine drawing any other line through this point. It can't be done without intersecting the graph at one other point. You can see this on the graph as well. There is a slider for the value k. This changes the slope of the line through the point (a, a3). When k = 1 it is the tangent line. For any other value the line intersects the parabola at two points (you may have to zoom waaaaay out to see this). There is one exception to this, but it will not concern us. Now we are going to find the equation for the tangent line to this graph at the point (2,4) algebraically. The process will also confirm our geometric intuition that there is only one linear function that intersects the graph only once at each point on the graph. You may know how to find this line with calculus if you have seen derivatives before. Do not use calculus to do this. Follow the method I outline below. We have plenty of time for calculus during the rest of the quarter! Method We want the slope of the tangent line to the graph of f(x) = x² at the point (2,4). We will need a point and the slope. We already have the point, so we need to find the slope. For the sake of tradition, let's call the slope m. • Write the equation of a line through the point (2, 4) with the slope m in point-slope form. • Rearrange this equation so that y is isolated. • Write an equation whose solutions would be the intersection of f(x) and this line. • When does this equation have only one solution? Hint: Quadratic Formula. Find this solution and write the equation of the tangent line. • Confirm your solution with graphing software like Desmos. Include an image or a link with your submission.