MHB Calculus FRQ: Implicit Curve, Slope & Vertical Tangents - Ethan

AI Thread Summary
The discussion focuses on solving a calculus problem involving the implicitly defined curve x^3 + y^3 = 4xy + 1. The general expression for the slope of the curve is derived as dy/dx = (4y - 3x^2) / (3y^2 - 4x). Vertical tangents occur where 3y^2 - 4x = 0, leading to the coordinates of points with vertical tangents being expressed in terms of cubic roots. At the specific point (2,1), the rate of change in the slope of the curve is calculated to be 8/5. The solution provides a comprehensive approach to implicit differentiation and the analysis of tangent behavior on the curve.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Calculus FRQ help! Given the curve x^3+y^3=4xy+1?


a. Write the general expression for the slope of the curve.
b. Find the coordinates of the point on the curve where the tangents are vertical.
c. At the point (2,1) find the rate of change in the slope of the curve with respect to x.

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
Hello Ethan,

We are given the implicitly defined curve:

$$x^3+y^3=4xy+1$$

a.) Implicitly differentiating with respect to $x$, we obtain:

$$3x^2+3y^2\frac{dy}{dx}=4\left(x\frac{dy}{dx}+y \right)+0$$

Arranging with all terms having $$\frac{dy}{dx}$$ as a factor on the left, and everything else on the right, we have:

$$3y^2\frac{dy}{dx}-4x\frac{dy}{dx}=4y-3x^2$$

Factor the left side:

$$\left(3y^2-4x \right)\frac{dy}{dx}=4y-3x^2$$

Dividing through by $$3y^2-4x$$ we find:

$$\frac{dy}{dx}=\frac{4y-3x^2}{3y^2-4x}$$

b.) The tangents are vertical where:

$$3y^2-4x=0\implies x=\frac{3}{4}y^2$$

Substituting for $x$ into the original curve, there results:

$$\left(\frac{3}{4}y^2 \right)^3+y^3=4\left(\frac{3}{4}y^2 \right)y+1$$

$$\frac{27}{64}y^6+y^3=3y^3+1$$

Arrange in standard polynomial form:

$$27y^6-128y^3-64=0$$

Applying the quadratic formula, we find:

$$y^3=\frac{8}{27}\left(8\pm\sqrt{91} \right)$$

Hence:

$$y=\frac{2}{3}\sqrt[3]{8\pm\sqrt{91}}$$

$$x=\frac{1}{2}\sqrt[3]{155\pm16\sqrt{91}}$$

Thus, the points having vertical tangents are:

$$(x,y)=\left(\frac{1}{2}\sqrt[3]{155\pm16\sqrt{91}},\frac{2}{3}\sqrt[3]{8\pm\sqrt{91}} \right)$$

c.) $$\left.\frac{dy}{dx} \right|_{(x,y)=(2,1}=\frac{4(1)-3(2)^2}{3(1)^2-4(2)}=\frac{8}{5}$$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top