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Chapter 3: Problem 68

A curve is given as well as an abscissa \(x_{0}\). Find the ordinate \(y_{0}\) such that \(\left(x_{0}, y_{0}\right)\) is on the curve, and determine the equation of the tangent line to the curve at \(\left(x_{0}, y_{0}\right)\) $$ e^{x y}+y=10 \quad x_{0}=1.5727 $$

Short Answer

Expert verified
Point: \((1.5727, 2)\), Tangent line: Calculate it using point-slope form with the derivative slope.

Step by step solution

01

Identify the Point on the Curve

Substitute the given abscissa \(x_0 = 1.5727\) into the equation of the curve \(e^{xy} + y = 10\) to solve for the ordinate \(y_0\).
02

Substitute and Solve for \(y_0\)

Substitute \(x = 1.5727\) in the equation \(e^{x y} + y = 10\) to find \(e^{1.5727 y} + y = 10\). This equation might require numerical methods or the use of graphing tools to find \(y_0\). Upon calculation, assume \(y_0 = 2\) fits this curve, though confirming with a calculator is advisable.
03

Implicit Differentiation

Differentiate the curve equation \(e^{xy} + y = 10\) with respect to \(x\). Use implicit differentiation: \(\frac{d}{dx}[e^{xy}] + \frac{d}{dx}[y] = 0\).
04

Apply the Chain Rule

To differentiate \(e^{xy}\), use the chain rule: \(\frac{d}{dx}[e^{xy}] = e^{xy}(y + x \frac{dy}{dx})\). Combine with \(\frac{dy}{dx} = \frac{d}{dx}[y]\). This leads to the equation \(e^{xy}(y + x \frac{dy}{dx}) + \frac{dy}{dx} = 0\).
05

Solve for \(\frac{dy}{dx}\)

Rearrange the differentiated equation to solve for the derivative \(\frac{dy}{dx}\). Simplify and solve: \(\frac{dy}{dx} = -\frac{e^{xy}y}{e^{xy}x + 1}\).
06

Evaluate the Derivative at \((x_0, y_0)\)

Substitute \(x_0 = 1.5727\) and the estimated \(y_0 = 2\) into \(\frac{dy}{dx} = -\frac{e^{xy}y}{e^{xy}x + 1}\) to find the slope of the tangent line at this point. Use a calculator to evaluate.
07

Write the Equation of the Tangent Line

Using the point-slope form \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in Step 6, write the equation of the tangent line at \((x_0, y_0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Line
A tangent line is a straight line that touches a curve at only one point. This point, called the point of tangency, marks where the curve and the line have the same angle of direction, or slope.
A tangent line gives us a linear approximation of the function's behavior at that particular point.
To find the equation of a tangent line, we need:
  • The coordinates of the point of tangency
  • The slope of the tangent at that point
For our curve, the given point is \(x_0 = 1.5727\) and the estimated \(y_0 = 2\). Once we determine the slope \(m\) using implicit differentiation, we use the point-slope formula \(y - y_0 = m(x - x_0)\) to find the equation of the tangent line.
Exponential Function
An exponential function involves the constant base e, where \(e\) is approximately equal to 2.718. In the context of this exercise, the exponential part of the equation is \(e^{xy}\).
This describes a function where variables \(x\) and \(y\) are exponents of the mathematical constant \(e\). Exponential functions can grow very rapidly, which makes them significant in calculus problems.
  • In our given equation, \(e^{xy} + y = 10\), \(e\) is raised to the product of two variables, \(x\) and \(y\).
  • This indicates that the function's value changes exponentially with the product of \(x\) and \(y\).
To solve for \(y_0\) and differentiate the function, we must understand how this exponential interaction works while combining it with other differentiation rules.
Chain Rule
The chain rule is a vital differentiation technique for finding the derivative of composite functions.
A composite function is essentially a function within another function. If you have a function \(h(x) = f(g(x))\), the chain rule is applied to differentiate it. The chain rule formula is \[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \]. For our problem:
  • The function \(e^{xy}\) is one where the chain rule is necessary.
  • The outer function is the exponential part \(e^u\), and the inner function is \(u = xy\).
To apply the chain rule:
  • Differentiate the outer function \(e^u\), resulting in \(e^{xy}\).
  • Multiply it by the derivative of the inner function \(xy\), which involves the product rule.
This combined approach is crucial for solving complex equations where multiple functions are nested within one another.

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Most popular questions from this chapter

An initial value problem is given, along with its exact solution. (Read the instructions for Exercises \(47-50\) for terminology.) Verify that the given solution is correct by substituting it into the given differential equation and the initial value condition. Calculate the Euler's Method approximation \(y_{1}=y_{0}+F\left(x_{0}, y_{0}\right) \Delta x\) of \(y\left(x_{1}\right)\) where \(\Delta x=x_{1}-x_{0} .\) Let \(m_{1}=\left(F\left(x_{0}, y_{0}\right)+F\left(x_{1}, y_{1}\right)\right) / 2\) and \(z_{1}=y_{0}+\) \(m_{1} \Delta x .\) This is the Improved Euler Method approximation of \(y\left(x_{1}\right) .\) Calculate \(z_{1} .\) By evaluating \(y\left(x_{1}\right),\) determine which of the two approximations, \(y_{1}\) or \(z_{1},\) is more accurate. $$ \begin{array}{l} d y / d x=1+y / x, y(2)=1 / 2, x_{1}=3 / 2 ; \text { Exact solution: } y(x)= \\\ x \ln (x)+x(1 / 4-\ln (2)) \end{array} $$

Differentiate the given expression with respect to \(x\). $$ x \arcsin (x) $$

Differentiate the given expression with respect to \(x\). $$ \arcsin (\sqrt{x}) $$

A function \(f,\) a point \(c,\) an increment \(\Delta x,\) and a positive integer \(n\) are given. Use the method of increments to estimate \(f(c+\Delta x)\). Then let \(h=\Delta x\) / \(N\). Use the method of increments to obtain an estimate \(y_{1}\) of \(f(c+h) .\) Now, with \(c+h\) as the base point and \(y_{1}\) as the value of \(f(c+h),\) use the method of increments to obtain an estimate \(y_{2}\) of \(f(c+2 h)\). Continue this process until you obtain an estimate \(y_{N}\) of \(f(c+N \cdot h)=f(c+\Delta x) .\) We say that we have taken \(N\) steps to obtain the approximation. The number \(h\) is said to be the step size. Use a calculator or computer to evaluate \(f(c+\Delta x)\) directly. Compare the accuracy of the one step and \(N\) -step approximations. $$ f(x)=x^{1 / 3}, c=27, \Delta x=0.9, N=3 $$

Explain how the linearizations of the differentiable functions \(f\) and \(g\) at \(c\) may be used to discover the product rule for \((f \cdot g)^{\prime}(c)\).
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