Question

Find the angle at which the ourves $$ 2 x=x^{4}-x y+x^{5}, \quad x^{4}+y^{4}+5 x=7 y $$ intersect at the origin.

Short answer

Based on the given curves, $$2 x=x^{4}-x y+x^{5}$$ and $$x^{4}+y^{4}+5 x=7 y$$, confirm if they intersect at the origin and calculate the angle of intersection between the curves at the origin.

Step-by-step solution

Check if the curves intersect at the origin

We are given the equations of the curves as: $$ 2 x=x^{4}-x y+x^{5}, \quad x^{4}+y^{4}+5 x=7 y $$ Now, plug in the origin \((x = 0, y = 0)\) into each equation: $$ 2(0)=(0)^{4}-(0)(0)+(0)^{5}, \quad (0)^{4}+(0)^{4}+5(0)=7(0) $$ $$ 0 = 0, \quad 0 = 0 $$ Since both equations are true, we can conclude that the curves intersect at the origin.

Compute the derivatives

To calculate the slope of the tangent, we need to find the derivatives of the two curves with respect to \(x\) and \(y\). We'll treat \(x\) and \(y\) as independent variables and differentiate with respect to \(x\) to find \(\frac{dy}{dx}\). For the first curve \(2x = x^4 - xy + x^5\), differentiate with respect to \(x\): $$ \frac{dy}{dx} + \frac{d(2x)}{dx} - \frac{d(x^4)}{dx} + \frac{d(xy)}{dx} - \frac{d(x^5)}{dx} = 0 $$ $$ \frac{dy}{dx} = -2 + 4x^3 - y + 5x^4 $$ For the second curve \(x^4 + y^4 + 5x = 7y\), differentiate with respect to \(x\): $$ \frac{dy}{dx} - \frac{d(x^4)}{dx} + \frac{d(y^4)}{dx} - \frac{d(5x)}{dx} = \frac{d(7y)}{dx} $$ $$ \frac{dy}{dx} = -4x^3 - 5 $$ Step 3: Find the slopes at the origin Now, plug in the origin \((x = 0, y = 0)\) into the derivatives to find the slopes: $$ m_1 = -2 + 4(0)^3 - (0) + 5(0)^4 = -2 $$ $$ m_2 = -4(0)^3 - 5 = -5 $$

Calculate the angle of intersection

Now, use the tangent formula to find the angle between the slopes: $$ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1m_2} = \frac{-2 - (-5)}{1 + (-2)(-5)} = \frac{3}{11} $$ Now, calculate the angle \(\theta\) using the inverse tangent function: $$ \theta = \arctan\left(\frac{3}{11}\right) $$ So, the angle at which the curves intersect at the origin is approximately \(\arctan\left(\frac{3}{11}\right)\).

Key concepts

Understanding Derivatives

When analyzing the intersection of curves, derivatives play a crucial role. They help us determine the slope of the tangent lines at a specific point on each curve. The slope is vital in finding out the angle at which the curves intersect.
In our exercise, we need to differentiate the given curve equations to find their derivatives. Derivation involves taking each term related to the variable and applying rules to determine how the function changes at different points. For instance, the derivative of a constant is zero, and the derivative of a term with a variable is determined by lowering the power and multiplying by the coefficient.
The expression we get after differentiating lets us find the rate of change of the curve, hence determining the slope of the tangent line at the point of intersection.

• Key steps in finding derivatives include using the power rule and differentiating each term individually.
• The derivative gives a new function representing the slope of the tangent line for any point on the curve.

The Tangent Line

A tangent line is a straight line that touches a curve at one specific point, matching the curve's slope at that point. It is crucial for studying the way in which curves intersect.
At the point of intersection, the slope of these tangent lines gives an insight into the behavior of each curve. By finding these slopes, we can calculate the angle between the curves.
When we differentiate a curve to find its derivative, we are essentially determining the equation of the tangent line at any given point.

• The tangent line at the intersection tells us about the instantaneous rate of change at that point.
• Using the slopes of these tangent lines helps in finding the intersection angle between two curves.

Calculating the Angle of Intersection

To find where two curves intersect, calculating the angle between them provides deeper insight into their relationship. This involves using the slopes of the tangent lines at the intersection.
The formula to find the angle between two intersecting curves involves using the tangent function. Specifically, if we know the slopes of tangent lines, we use the formula:
\[\tan(\theta) = \frac{m_1 - m_2}{1 + m_1m_2}\]
where \(m_1\) and \(m_2\) are the slopes of the tangent lines derived from the derivatives of the curves.

• The angle \(\theta\) is found using the arctan or inverse tangent function.
• This calculation provides the intersection angle in terms of radians, which can often be converted into degrees if needed.

Differentiation Techniques

Differentiation is the mathematical process of finding a derivative. It is key in determining how a function behaves at any point and is crucial in finding the slopes of tangent lines.
In our case, differentiating functions involves several techniques such as:

• Power Rule: Used to find the derivative of a term with a power, such as \(x^n\) becomes \(nx^{n-1}\).
• Product Rule: Useful when taking the derivative of a product of two functions.
• Chain Rule: Helpful when dealing with composite functions.

These techniques allow us to handle complex curve equations and simplify them to find the derivatives. Differentiation helps to understand how fast or slow a curve is changing at any point on its path.

Algebraic Curves

Algebraic curves are equations formed by polynomials which represent geometric shapes on a coordinate plane. Understanding their behavior is important in various fields such as physics and engineering.
These curves can intersect at different angles depending on their shapes and equations. The study of their intersections involves analyzing the points where their equations hold true simultaneously, like in our given exercise.

• Algebraic curves can have different shapes such as lines, parabolas, hyperbolas, etc.
• The intersection points of these curves can be found by solving their equations simultaneously.
• Analyzing intersections helps in understanding relationships and predictions between different sets of data.

Overall, algebraic curves provide a visual and analytical way to understand mathematical relations and help in solving real-world problems.