Solving Two Mathematical Problems: Graphing & Bisection Method

a. To sketch the graphs of y = e^x - 2 and y = cos(e^x - 2), let's analyze each equation separately.

For the equation y = e^x - 2:

1. Determine key points:
   • When x = 0, y = e^0 - 2 = 1 - 2 = -1.
   • As x approaches negative infinity, y approaches -2.
   • As x approaches positive infinity, y approaches positive infinity.
2. Plot the points and draw the curve:
   • The graph will start below the x-axis, pass through the point (0, -1), and then continuously increase as x increases, approaching positive infinity.

For the equation y = cos(e^x - 2):

1. Determine key points:
   • When x = 0, y = cos(e^0 - 2) = cos(1 - 2) = cos(-1) = 0.54 (approximately).
   • As x approaches negative infinity, y oscillates between -1 and 1.
   • As x approaches positive infinity, y oscillates between -1 and 1.
2. Plot the points and draw the curve:
   • The graph will oscillate above and below the x-axis, passing through the point (0, 0.54).

b. To find an approximation to within 10^-5 for the equation e^x - 2 = cos(e^x - 2) using the Bisection method on the interval [0.5, 1.5], follow these steps:

1. Evaluate the function at the endpoints of the interval:

   • For x = 0.5, compute f(0.5) = e^(0.5) - 2 - cos(e^(0.5) - 2).
   • For x = 1.5, compute f(1.5) = e^(1.5) - 2 - cos(e^(1.5) - 2).
   • Check if the product of f(0.5) and f(1.5) is negative. If not, choose a different interval.

2. Divide the interval in half and evaluate the function at the midpoint:

   • For the interval [0.5, 1.5], the midpoint is x = (0.5 + 1.5) / 2 = 1.
   • Compute f(1) = e^1 - 2 - cos(e^1 - 2).

3. Determine the subinterval that contains a root:

   • If f(1) is very close to zero (within the desired tolerance), then x = 1 is an approximation.
   • Otherwise, determine which half of the interval (either [0.5, 1] or [1, 1.5]) contains a root based on the signs of f(0.5) and f(1).

4. Repeat steps 2 and 3 with the selected subinterval until the approximation is within the desired tolerance.

Using this method, you can iteratively refine the interval and find an approximation to within 10^-5 for the equation e^x - 2 = cos(e^x - 2).