Graphical solution of pure-time differential equations

Math 201, Spring 2015
Name:
ID #:

Due date:
Table/group #:
Group members:

Our goal is to understand what a pure-time differential equation can tell us about the graph of the unknown function.
1. Recall using the slope of the tangent line to find the derivative of a function at a point. If a function is decreasing, the slope is negative, so the derivative is negative. Likewise, if the derivative is negative at a point, the function is
  there.
2. If a function is increasing, the slope is positive, so the derivative is positive. Likewise, if the derivative is positive at a point, the function is
  there.
3. If the function is flat, or has a horizontal tangent line, the slope and derivative are zero. Then if the derivative is zero at a point, the function is
  there. In other words, the graph is
  .
Let's use our understanding of what the derivative tells us about the shape of a graph to sketch the solution to the following pure-time differential equation: dwdt=f(t)w(0)=−2
Here's a graph of f.

The function $w(t)$ is the solution to the differential equation. We don't yet know what the function $w(t)$ is, but we do know that at each time point $t$ , its derivative must be equal to the value of $f$ at time $t$ , as graphed above.
1. For what values of $t$ is the graph of $w$ horizontal? (If there are multiple, enter them in increasing order, separated by commas.)
  
  For what values of $t$ is $w$ increasing?
  
  For what values of $t$ is $w$ decreasing?
  
  Need help?Hide help
  
  Hint
  If the graph of $w$ is horizontal, then what must its derivative be? (See previous problem.) Since the derivative of $w$ is $f$ , you just need to find times $t$ where $f$ is that value.
  If the graph of $w$ is increasing, then what must its derivative be? Find times $t$ where its derivative, $f$ , satisfies that condition.
  
  If the graph of $w$ is decreasing, then what must its derivative be? Find times $t$ where its derivative, $f$ , satisfies that condition.
  
  For the questions about increasing and decreasing, use inequalities to represent the valid values of $t$ . For example, if you determined that $w(t)$ is decreasing for all times after $t=6$ , then your answer would be t > 6
  Hide help
2. Our next goal is to sketch what the solution $w(t)$ looks like.
  
  First, identify the initial condition: $w($ $)=$
  . Start your sketch of the solution by plotting this point.
  
  Feedback from applet
  initial condition:
  peak location:
  shape:
  (You'll finish graphing the solution on the above axes in part c.)
  Need help?Hide help
  
  Hint
  In the applet, use the left of the two points to graph the initial condition. You won't be able to get full credit from the applet until finishing part c, below.
  Hide help
3. Next, use your results from part a to determine whether $w$ starts off by increasing or decreasing. Since when $t=0$ , $w$ is , we need the graph the of $w(t)$ to be the initial condition for values of $t$ just above $0$ .
  
  However, $w(t)$ does not keep moving in the same direction. Instead, it stops and reverses direction at $t=$ . Make sure your sketch accurately represents what happens to $w(t)$ at this value of $t$ .
  
  Need help?Hide help
  
  Hint
  In the applet, you can move the point on the right up or down to achieve the correct choice of increasing or decreasing, and left or right to shift the vertex of the parabola to the correct value of $t$ .
  Since we don't give you a formula for the actual function $f$ , there is not a single correct answer for the solution $w(t)$ . Any graph that show the correct qualitative behavior is fine.
  
  Hide help
Let's use our understanding of what the derivative tells us about the shape of a graph to sketch the solution to the following pure-time differential equation, where f is graphed below: dwdt=f(t)w(0)=5
1. For what values of $t$ is the graph of $w$ horizontal? (If there are multiple, enter them in increasing order, separated by commas.)
  
  For what values of $t$ is $w$ increasing? It is increasing for
  .
  
  For what values of $t$ is $w$ decreasing? It is decreasing for
  .
  
  Need help?Hide help
  
  Hint
  Remember, we are not asking where the graph of $f$ is horizontal, increasing, or decreasing. Instead, we are asking where the graph of the solution $w(t)$ is horizontal, increasing, or decreasing, just like we did, above.
  If, for one of the answers, you want say $t$ is between two numbers, i.e., $a < t < b$ , you can write your answer in one of two ways. You can use interval notation $t \in (a,b)$ , where the symbol $\in$ means “is an element of.” Online, you can type this as t in (a,b). Alternatively, you can write your answer as t > a and t < b.
  
  If, on the other hand, you want to let one of the bounds be infinity, use inequalities. Multiple inequalities can be combined by typing or between them (e.g., t < c or t > d).
  
  Hide help
2. Now we will sketch the solution on the below axes.
  
  First, identify the initial condition: $w($ $)=$
  . If you are sketching by hand, you'll want to start your sketch of the solution by plotting this point. In the below applet, you will have to adjust other points later to make sure the initial condition is satisfied.
  
  Next, use your results from part a to determine whether $w$ begins by increasing or decreasing. Since when $t=0$ , $w(t)$ is , we need the graph to be the initial condition for values of $t$ just above $0$ .
  
  From the above graph of the derivative of $w(t)$ (i.e, the graph of $f$ ), you can determine that $w(t)$ stops and changes directions at $t=$ . (If there are multiple values, enter them in increasing order, separated by commas.) Make sure your sketch of the solution $w(t)$ changes direction at these values of $t$ to reflect this reality. At each point, determine whether the graph should have a local maximum, minimum, or neither. Enter maximum, minimum, or neither, separated by commas, in the same order as the corresponding values of $t$ :
  
  For the sketch to be considered accurate, it must match the initial condition, be increasing and decreasing over the correct intervals, and have local maxima and minima at the correct values of $t$ . Since we don't have a formula for $f$ , we aren't concerned with the precise values of $w(t)$ .
  
  Feedback from applet
  initial condition:
  peak locations:
  shape:
  
  Need help?Hide help
  
  Hint
  
  To sketch the solution $w(t)$ with the applet, first move the points left or right until they are at the correct values of $t$ where $w(t)$ changes direction. Move the points up or down to get the correct shape, with the right points being local maxima or local minima. Lastly, further adjust one or both points up or down until the graph satisfies the initial condition.
  
  In the applet, which of the two points is a maximum and which is a minimum depends only on which one is higher.
  
  Hide help

Math Insight

Graphical solution of pure-time differential equations

Hint

Hint

Hint

Hint

Hint

Thread navigation

Math 201, Spring 2015