# Write a system of equations given the graph of the derivative

We can also convert the initial conditions over to the new functions. Remember that in rationalizing the numerator in this case we multiply both the numerator and denominator by the numerator except we change the sign between the two terms.

It makes sense that the number of prey present will affect the number of the predator present. In fact, we are going to see that there are only a few simple steps for writing the equation of a tangent line or normal line perpendicular line to a curve at a given point.

Developing an effective predator-prey system of differential equations is not the subject of this chapter. Systems of differential equations can be converted to matrix form and this is the form that we usually use in solving systems.

Example 1 Find the derivative of the following function using the definition of the derivative. This will lead to two differential equations that must be solved simultaneously in order to determine the population of the prey and the predator.

You do remember rationalization from an Algebra class right? Here are the steps: The next theorem shows us a very nice relationship between functions that are continuous and those that are differentiable.

Here is an example of a system of first order, linear differential equations.

Also note that we wrote the fraction a much more compact manner to help us with the work. So, we are going to have to do some work. Example 1 Write the following 2nd order differential equation as a system of first order, linear differential equations.

Substitute the given x-value into the function to find the y-value or point. So, cancel the h and evaluate the limit. So, upon canceling the h we can evaluate the limit and get the derivative.

Here is the official definition of the derivative. Why is this fascinating?As wikiHow, nicely explains, to find the equation of a line tangent to a curve at a certain point, you have to find the slope of the curve at that point, which requires calculus.

Meaning, we need to find the first derivative. Here are the steps: Substitute the given x-value into the function to find the y-value or point.

The equation of the tangent at x = 1 has slope 8 and passes through (1, 0) and its equation is given by: y = 8x - 8 The equation of the tangent at x = -2 has slope -4 and passes through (-2, ) and its equation is given by: y = - 4x - 14 c) Graphs of the quadratic function and all three tangent lines.

Figure 1. However, systems can arise from $$n^{\text{th}}$$ order linear differential equations as well. Before we get into this however, let’s write down a system and get some terminology out of the way. We are going to be looking at first order, linear systems of differential equations.

These terms mean the same thing that they have meant up to this point. Functions & Graphing. Line Equations Functions Arithmetic & Comp. Conic Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp Find the value of a function derivative at a given point.

Derivatives. First Derivative. Looking at the graph of the derivative in the x,y-plane it is easy to very determine the important information. Here is a summary relating the features of the graph of the derivative with the graph of the function.

Writing systems of equations that represents the charges by: Anonymous Jenny charges $4 per day to pet sit. Tyler charges$2 up front, and then \$3 per day to pet sit. Write a system of equations that represents the charges.

_____ Your answer by Karin from ultimedescente.com: You want to write two equations.

Write a system of equations given the graph of the derivative
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