CoCalc -- Implicit Differentiation Assignment.sagews

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Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

For Questions 1-3, perform each of the following steps (follow Example 7).

Calculate the derivative $\frac{dy}{dx}$ , and plug in the given $x$ - and $y$ -values to get the slope, $m$ .
Calculate the tangent line at the given point $(x_0,y_0)$ : $TL(x)=y_0+m\cdot(x-x_0)$ .
Graph the implicit function and the tangent line on the same window.

$y^4-4y^2-x^4+9x^2=0;\quad (0.5888,1)$

$x^3+y^3=9xy;\quad (2,4)$

$\displaystyle (x^2+y^2-1)^3=x^2y^3;\quad (1,1)$

Consider the curves $y^2=x^3$ and $2x^2+3y^2=5$ .

Find $\frac{dy}{dx}$ for the first curve.

Find the tangent line to the first curve at $(1,1)$ .

Find $\frac{dy}{dx}$ for the second curve.

Find the tangent line to the second curve at $(1,1)$ .

[Make sure you give this tangent line a different name than the tangent line in Part b.]

Graph the two curves and the two tangent lines on the same axes (use red for the tangent lines).

[Notice that the two tangent lines are perpendicular.]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is