Finding Tangent Lines and Intersections: A Practice Guide for Calculus Final

[updayte]

Hello, I am preparing for my Calc final and have these a ton of practice questions, these are a few I am not getting. Can somebody help me out with these?

1. find the tangent line slope of
y=g(x)=1/2x+1 at x =-2

2. Derivative of y with respect to x
a) y=(3x+4/2x-4)^1/10
b) find dy/dx if y²=sin(xy)

3. What value of k is the line y=kx tangent to the curve y= 4x²-x³

 

[mutton]

1. The slope of the line tangent to a differentiable function at a point is the derivative at that point.

2.
a) This is a composite function of x, so use the chain rule.
b) It's hard to isolate for y, so use implicit differentiation.

3. The slope of the line and the curve at any point can be found by differentiation. Also notice that the line passes through (0, 0). Another way of phrasing this question is: What is the slope of the line passing through (0, 0) tangent to the curve? See https://www.physicsforums.com/showthread.php?t=284498 part (d) for a similar question.

 

[sennyk]

Problem 3 is fun! There are two solutions for k.

Create a model that will tell where the equations intersect.
Then create a model that will tell when their slopes are the same.

You should end up with two equations in terms of k and x.