Problem 60 - University Calculus: Early Transcendentals

Question

Use a CAS to perform the following steps a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point \(P\) satisfies the equation. b. Using implicit differentiation, find a formula for the derivative \(d y / d x\) and evaluate it at the given point \(P\) c. Use the slope found in part (b) to find an equation for the tangent line to the curve at \(P .\) Then plot the implicit curve and tangent line together on a single graph. $$y^{3}+\cos x y=x^{2}, \quad P(1,0)$$

Step-by-Step Solution

Answer

Point P satisfies the curve, slope at P is 2, tangent line equation: \(y=2x-2\).

Step 1: Implicit Plot

Use a CAS (Computer Algebra System) to plot the equation \(y^3 + \cos(xy) = x^2\). Plot this equation and visually inspect whether the point \(P(1,0)\) lies on the curve. You will see that for this point, substituting \(x=1\) and \(y=0\) satisfies the equation: \(0^3 + \cos(1 \cdot 0) = 1^2\), so the point \(P(1,0)\) is on the curve.

Step 2: Implicit Differentiation

Differentiate the equation \(y^3 + \cos(xy) = x^2\) implicitly with respect to \(x\). Use the chain rule for \(\cos(xy)\) and apply the product rule implicitly. After differentiating, you will have:\[ 3y^2 \frac{dy}{dx} - \sin(xy)\left(y + x\frac{dy}{dx}\right) = 2x \]Rearrange to solve for \(\frac{dy}{dx}\).

Step 3: Solve for the Derivative

Rearranging the differentiated expression gives:\[ \frac{dy}{dx} = \frac{2x + y \sin(xy)}{3y^2 + x \sin(xy)} \]Substitute the point \(P(1,0)\) into this derivative formula:At \((x, y) = (1, 0)\), the derivative simplifies to \(\frac{dy}{dx} = 2\).

Step 4: Equation of the Tangent Line

Use the point-slope form of a line equation to find the tangent line at point \(P(1,0)\), given the slope \(m = 2\). The equation of the tangent line is:\[ y - 0 = 2(x - 1) \]Simplify this to get the tangent line equation: \(y = 2x - 2\).

Step 5: Plot Curve and Tangent Line

Plot both the original implicit curve \(y^3 + \cos(xy) = x^2\) and the tangent line equation \(y = 2x - 2\) on the same graph using a CAS. Verify visually that the tangent line is correctly positioned and tangent at the point \(P(1,0)\).

Key Concepts

Computer Algebra System (CAS)

When dealing with complex mathematical equations, a Computer Algebra System (CAS) becomes a powerful tool.CAS tools can handle algebraic expressions, perform symbolic computations, and plot complex graphs.They are particularly useful for tasks like implicit differentiation and plotting equations that are not readily solvable by hand.
In our exercise, the CAS is employed to plot the implicit equation \(y^3 + \cos(xy) = x^2\). This type of mathematical software can showcase the otherwise invisible relationships between variables.By visually representing the equation on a graphical interface, students can verify solutions through observation.
One significant advantage of using a CAS is its ability to check whether a given point, such as \(P(1,0)\), lies on the plotted curve. This is done by substituting the coordinates into the equation and confirming if it holds true.Through such computational verification, students can improve their understanding of how points, lines, and curves interconnect.

Tangent Line

A tangent line to a curve at a given point is a straight line that just touches the curve at that point without crossing it.It represents the instantaneous rate of change (or slope) of the curve at that point.
To find the tangent line to a curve, you start by calculating the derivative of the function, which provides the slope of the tangent line.In our example, implicit differentiation is used to determine the slope of the curve at point \(P(1,0)\). This derivative gives us a slope of 2 at that point.
With this slope, the equation for the tangent line can be found using the point-slope formula.For a point \((x_0, y_0)\) with slope \(m\), the formula is \(y - y_0 = m(x - x_0)\).Substituting our known values gives us the tangent line equation \(y = 2x - 2\).This precise tangent line can then be plotted alongside the original curve to visually confirm its accuracy.

Implicit Plotting

Implicit plotting is a method to graph equations that are not explicitly solved for one variable in terms of another.Unlike explicit functions expressed as \(y = f(x)\), implicit equations entwine variables in a more complex manner.
In the equation \(y^3 + \cos(xy) = x^2\), neither variable is isolated, demonstrating the need for implicit plotting.Using a CAS, you can input this equation and receive a graphical depiction of its curve.
This visualization helps in understanding the shape and intersections of graphs, enabling students to see abstract relationships more concretely.Implicit plotting is essential when analyzing complex curves, as it provides insight into points of intersection, such as verifying if \(P(1,0)\) is on the curve.By plotting the tangent line together with the implicit curve, you gain a comprehensive graphical representation, assisting in grasping the mathematical concepts more fully.