Subject: Calculus AB AP --tangent lines
Name: Melissa
Who are you: Student

let f be a function with f(1)=4 such that for all points (x,y) on the graph of f the slope is given by (3x^(2)+1)/(2y)
a.)Find the slope of the graph of f at the point where x=1.
b.)Write an equation for the line tangent to the graph of f at x=1 and use it to approximate f(1.2)
c.) Find whether f is concave up or concave down when x=1. Is your answer in part b an overestimate or an underestimate?

Hello Melissa.

1. You are given a function of x and y that returns the slope at that point. You also know that f(1) = 4, so that means (1,4) are the values of (x,y) to put into the slope function. Just plug them in and get the slope.
   
   
2. From part (a) you have the slope of the graph at the point (1,4). That's the tangent line, so use this point and the slope to write an equation of the tangent line here. That's (y-y0) = m(x-x0), you will recall. Then estimate f(1.2) by
   putting x=1.2 into your tangent line equation and calculate y.
   
   
3. Take the second derivative of the function by taking the derivative of the slope function you were given. You'll need to use the chain rule, since "y" is in there, but remember that dy/dx *is* the slope function you were given earlier. It's a fair bit of algebra to sort it all out, but when you have done the differentiation, you can plug in (again) x=1 and y=4 and find out if the second derivative is positive or negative, which tells you the concavity. The concavity will tell you whether your estimate is an underestimate or an overestimate. You can see this if you draw any concave up curve, then draw a tangent line somewhere to this curve and see if the line is above (overestimate) or below (underestimate); a concave down curve is similar.

Hope this helps,
Stephen La Rocque.