Question

Please see attached.

Answer 1

`y=-(17)/(6)x+(35)/(6)`

Explanation:

To find the equation of the tangent line to the curve at the specified point, we first need to understand what a tangent line is and how to find it.

Tangent Line:

A tangent line to a curve is a straight line that touches the curve at a single point and has the same slope as the curve at that point. In other words, it "just grazes" the curve at that particular spot without crossing it. The tangent line represents the instantaneous rate of change of the curve at that specific point.

Derivative:

To find the slope of the tangent line at a given point on the curve, we need to calculate the derivative of the curve at that point. The derivative of a function with respect to a variable measures how the function changes concerning that variable. In our case, we'll need to find the derivative with respect to the variable "x".

Steps to Find the Equation of the Tangent Line:

1. Take the derivative of the given curve with respect to 'x' to find the slope of the tangent line at any point (x, y) on the curve.
2. Substitute the coordinates of the specified point P(x₀, y₀) into the derivative to find the slope of the tangent line at that point.
3. Use the point-slope form of a line (y - y₀ = m(x - x₀)) with the slope and the point (x₀, y₀) to get the equation of the tangent line.

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Applying these steps to the given curve 2x⁴ + xy² = 11 at the point P(1, 3):

Step 1: Find the derivative with respect to "x":

Differentiating the given equation implicitly with respect to "x."

`\Longrightarrow (d)/(dx) [2x^4 + xy^2] = (d)/(dx) [11]`

Using the product rule and chain rule for differentiation, we get:

`\Longrightarrow \boxed{8x^3 + y^2+ 2xy\Big((dy)/(dx) \Big) = 0}`

Step 2: Substitute the coordinates of point P(1, 3) into the derivative:

We already know x = 1 and y = 3. Now, we need to find dy/dx at this point.

`\Longrightarrow 8(1)^3+(3)^2 + 2(1)(3)\Big((dy)/(dx) \Big) = 0`

Solving for dy/dx:

`\Longrightarrow 8+9 + 6\Big((dy)/(dx) \Big) = 0\\\\\\\\\Longrightarrow 17+6\Big((dy)/(dx) \Big) = 0\\\\\\\\\Longrightarrow 6\Big((dy)/(dx) \Big) = -17\\\\\\\\\therefore (dy)/(dx) \Big = -(17)/(6)`

Step 3: Use the point-slope form to write the equation of the tangent line. The point-slope form of the line is given by:

y - y₀ = m(x - x₀)

Where m is the slope we calculated in Step 2 (dy/dx) and (x₀, y₀) are the coordinates of point P(1, 3).

Substituting the values:

`\Longrightarrow y - 3 = -(17)/(6) (x - 1)\\\\\\\\\Longrightarrow y - 3 = -(17)/(6)x + (17)/(6)`

Now, if we want the equation in slope-intercept form (y = mx + b), we can rearrange the terms:

`\Longrightarrow y = -(17)/(6)x + (17)/(6)+3\\\\\\\\\therefore \boxed{\boxed{y=-(17)/(6)x+(35)/(6) }}`

So, the equation of the tangent line to the curve 2x⁴ + xy² = 11 at the point P(1, 3) is y = (-17/6)x + 35/6.

`\hrulefill`

Additional Terminology:

Implicit Differentiation:

Implicit differentiation is a technique used to find the derivative of an equation where the dependent and independent variables are not explicitly separated. In this method, you differentiate both sides of the equation with respect to the variable of interest and then solve for the derivative.

Point-Slope Form of a Line:

The point-slope form of a linear equation is y - y₀ = m(x - x₀), where m is the slope of the line, and (x₀, y₀) are the coordinates of a point on the line. This form is useful when you know a point on the line and its slope and want to write the equation of the line.

Slope-Intercept Form of a Line:

The slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope of the line, and 'b' represents the y-intercept, which is the point where the line intersects the y-axis.

Chain Rule:

The chain rule is a formula used to find the derivative of a composite function. If you have a function that is a composition of two or more functions, the chain rule allows you to find the derivative by taking the derivative of the outer function and then multiplying it by the derivative of the inner function.

`\boxed{\left\begin{array}{ccc}\text{\underline{Chain Rule:}}\\\\(d)/(dx)[f(g(x))]=f'(g(x)) \cdot g'(x) \end{array}\right }`

Product Rule:

The product rule is a formula used to find the derivative of a product of two functions. If you have a function that is the product of two functions f(x) and g(x), the product rule states that the derivative of the product f(x) · g(x) with respect to x is given by:

`\boxed{\left\begin{array}{ccc}\text{\underline{Product Rule:}}\\\\(d)/(dx)[f(x)g(x)]=f(x)g'(x)+g(x)f'(x) \end{array}\right }`