Skills Review for Direction Fields and Numerical Methods

Learning Outcomes

  • Calculate the slope of a tangent line
  • Solve polynomial equations

In the Direction Fields and Numerical Methods section, we will need skills including how to find the slope of a tangent line and solve polynomial equations. These skills are reviewed here.

Calculate the Slope of a Tangent Line

Recall that a derivative can be used to find the slope of tangent line at a specific point.

Example: Finding the Slope of a Tangent Line

Find the slope of the line tangent to the graph of [latex]f(x)=x^2-4x+6[/latex] at [latex]x=1[/latex].

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Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[/latex] at [latex]x=2[/latex]. Use the point-slope form.

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Example: Finding The Slope of a Tangent Line

Find the slope of the line tangent to the curve [latex]x^2+y^2=25[/latex] at the point [latex](3,-4)[/latex]. Note the derivative of the equation is [latex]\frac{dy}{dx}=-\frac{x}{y}[/latex].

Example: Finding the Slope of the Tangent Line

Find the slope of the line tangent to the graph of [latex]y^3+x^3-3xy=0[/latex] at the point [latex]\left(\frac{3}{2},\frac{3}{2}\right)[/latex]. Note the derivative of the equation is [latex]\frac{dy}{dx}=\frac{3y-3x^2}{3y^2-3x}[/latex].

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Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[/latex] at the point [latex](5,3)[/latex]. Note the derivative of the equation is [latex]\frac{dy}{dx}=\frac{x}{y}[/latex].

Solve Polynomial Equations

A General Note: Polynomial Equations

A polynomial of degree n is an expression of the type

[latex]{a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\cdot \cdot \cdot +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

where n is a positive integer and [latex]{a}_{n},\dots ,{a}_{0}[/latex] are real numbers and [latex]{a}_{n}\ne 0[/latex].

Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent n.

Example: Solving a Polynomial Equation

Solve the polynomial equation [latex](x-2)^2(x^2+5x+6)=0[/latex].

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