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 $x=1$ .

Show Solution

Since the slope of the tangent line at 1 is $f^{\prime}(1)$ , we must find $f^{\prime}(x)$ . Using the definition of a derivative, we have

f^{'} (x) = 2 x - 4

$f^{\prime}(x)=2x-4$

so the slope of the tangent line is $f^{\prime}(1)=-2$ .

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.

Show Solution

$m=12$

Try It

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 $\frac{dy}{dx}=-\frac{x}{y}$ .

Show Solution

The slope of the tangent line is found by substituting $(3,-4)$ into the derivative. Consequently, the slope of the tangent line is $\frac{dy}{dx}|_{(3,-4)} =-\frac{3}{-4}=\frac{3}{4}$ .

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 $\frac{dy}{dx}=\frac{3y-3x^2}{3y^2-3x}$ .

Show Solution

Substitute $(\frac{3}{2},\frac{3}{2})$ into $\frac{dy}{dx}=\frac{3y-3x^2}{3y^2-3x}$ to find the slope of the tangent line:

\frac{d y}{d x} |_{(\frac{3}{2}, \frac{3}{2})} = - 1

$\frac{dy}{dx}|_{(\frac{3}{2},\frac{3}{2})}=-1$

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 $\frac{dy}{dx}=\frac{x}{y}$ .

Show Solution

$\frac{5}{3}$

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}

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

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

Show Solution

Since the equation is already set equal to 0, begin by factoring the equation entirely:

$(x-2)^2(x+3)(x+2)=0$

Now, set each factor equal to 0.

$(x-2)^2=0$ yields $x=2$

$x+3=0$ yields $x=-3$

$x+2=0$ yields $x=-2$

The solutions are $x=-3,-2,2$

Try It

Hint

Differential Equations: Prerequisite Material