Use a graph to find a number delta such that if abs(x-pi/4) < delta then abs(tanx-1) < 0.2. I understand that the epsilon is 0.2. But what step do I take next? How do I solve the problem?
To solve the problem, you need to graph the functions |x - pi/4| and |tan x - 1|. Here's a step-by-step approach to find the value of delta:
1. Plot the graph of the function f(x) = |x - pi/4|. This graph will be a symmetric V-shape, centered around x = pi/4. The vertex of the V-shape will be at (pi/4, 0). Note that the function f(x) represents the distance between x and pi/4.
2. Next, plot the graph of the function g(x) = |tan x - 1|. This graph will be a wavy curve. Note that the function g(x) represents the distance between tan(x) and 1.
3. Now, you want to find a value of delta such that if |x - pi/4| < delta, then |tan x - 1| < 0.2. In other words, you want to find the points on the graph of f(x) that lead to points on the graph of g(x) with a vertical distance less than 0.2.
4. Start with a small value of delta, let's say 0.1. Shade the region on the graph of f(x) where |x - pi/4| < 0.1.
5. Look at the corresponding region on the graph of g(x). If the shaded region on the graph of g(x) is entirely within a vertical distance of 0.2 from the x-axis (i.e., it doesn't exceed a height of 0.2), then you've found a valid value of delta. If not, adjust the value of delta and repeat steps 4 and 5 until you find the desired delta.
By following these steps and iteratively adjusting the value of delta, you can find a suitable value that satisfies the condition |x - pi/4| < delta and |tan x - 1| < 0.2.