Using a graphing calculator and the concept of differentials determine the values for x when 1/(1+3x)^5 is approx = to (1-15x) within .01
since the line 1-15x is the tangent line at (0,1), we could just find where
|1/(1+3x)^5) - (1-15x)| < 0.01
-0.008353 < x < 0.008872
Using differentials, we have
y(0) = 1
dy = -15/(1+3x)^6 dx
So, if we want |dy| < 0.01, we need x such that
1/(1+3x)^5 - (1-15x) < 0.01
which is just what we had above.
I need help please, how can I get a graphing calculator?
To determine the values of x when 1/(1+3x)^5 is approximately equal to (1-15x) within a deviation of 0.01, we can utilize a graphing calculator and the concept of differentials. Here's a step-by-step guide to finding these values:
1. Enter the function 1/(1+3x)^5 as Y1 in your graphing calculator.
2. Enter the function (1-15x) as Y2 in your graphing calculator.
3. Graph both functions by selecting "Graph" or a similar function on your calculator.
4. Look for the point of intersection on the graph. This is where the two functions are equal to each other.
5. Use the trace function or a similar tool on your calculator to find the x-coordinate of the point of intersection. This will give you the initial approximation for x when the functions are approximately equal.
6. To improve the accuracy of the approximation, we'll use differentials. Calculate the derivative of the function 1/(1+3x)^5 with respect to x. This will give you the differential equation.
7. Solve the differential equation for x to find the differential approximation for x.
8. Add the differential approximation to the initial approximation obtained from the graph to refine the approximation further.
9. Repeat steps 6-8 until the approximation for x is within the desired deviation of 0.01.
By following these steps, you can utilize a graphing calculator and the concept of differentials to determine the values for x when 1/(1+3x)^5 is within 0.01 of (1-15x).