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 $f(x)=x^2-4x+6$ at $x=1$.

Try It

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

Example: Finding The Slope of a Tangent Line

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

Example: Finding the Slope of the Tangent Line

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

Try It

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

Solve Polynomial Equations

A General Note: Polynomial Equations

A polynomial of degree n is an expression of the type

${a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\cdot \cdot \cdot +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

where n is a positive integer and ${a}_{n},\dots ,{a}_{0}$ are real numbers and ${a}_{n}\ne 0$.

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 $(x-2)^2(x^2+5x+6)=0$.