Slopes of Tangent Lines on a Curve
When we graph a differentiable function we have a tangent line at each point on that graph. Similarly, we may be able to find tangent lines at points on implicitly defined curves. As we saw in lecture, we can often assume that near a given point, the curve represents the graph of an explicit function \( y=y(x) \). We can then use implicit differentiation to compute \(dy/dx\), which represents the slope of the tangent line to the curve. Exercise 3 below and Exercise 4 on the following page provide interactive tools to explore this idea.
3. Consider the curve given by your instructor. Use the three SageMath cells below to answer the following questions. Record your answers on your lab report.
(a) By replacing the “Equation to solve” appropriately in the first SageMath cell below, find the points on the curve where \( x \) equals the value given in your lab report.
(b) For each of the points you found in (a), determine the slope of the tangent line to the curve at that point. The second SageMath cell below can be used to compute an expression for \({dy}/{dx}\) and the third SageMath cell below can be used to compute the slope given \( dy/dx \) and the point.
(c) For each of the points you found in (a), write an equation for the tangent line to the curve at that point.