Let f x coth x . Use the Graph software posted in Moodle to do the following:
(a) Graph f x .
(b) Graph the tangent and normal lines to the graph of f x at x 1.
(c) Graph f x .
(d) From the graphs obtained in parts (a) – (c), answer the following questions
(i) lim
x
f x
(ii) lim
x
f x
(iii)
0
lim
x
f x
(iv)
0
lim
x
f x
(v) explain why f x is not differentiable at x 0
better try just typing in the text, rather than copying from you application.
To answer these questions, you can use graphing software to plot the function and analyze its behavior. Here's how you can proceed step by step:
(a) Graphing f(x):
1. Open the graphing software posted in Moodle.
2. Enter the function f(x) = coth(x).
3. Set the range for x-values you want to display on the graph.
4. Click the "Graph" or "Plot" button to generate the graph of f(x).
(b) Graphing tangent and normal lines:
1. Identify the point on the graph where x = 1.
2. Calculate the derivative of f(x), denoted as f'(x).
3. Calculate the slope of the tangent line at x = 1 using the derivative.
4. Calculate the slope of the normal line at x = 1 using the negative reciprocal of the tangent line's slope.
5. Use the slope and the point (1, f(1)) to find the equation of the tangent line.
6. Use the slope and the point (1, f(1)) to find the equation of the normal line.
7. Plot the tangent and normal lines on the graph of f(x) at x = 1.
(c) Graphing f'(x):
1. Calculate the derivative of f(x), denoted as f'(x).
2. Enter the function f'(x) into the graphing software.
3. Set the range for x-values you want to display on the graph.
4. Click the "Graph" or "Plot" button to generate the graph of f'(x).
(d) Analyzing the graphs:
(i) To find lim(x->-∞) f(x), observe the behavior of f(x) as x approaches negative infinity on the graph.
(ii) To find lim(x->∞) f(x), observe the behavior of f(x) as x approaches positive infinity on the graph.
(iii) To find lim(x->0-) f(x), observe the behavior of f(x) as x approaches 0 from the negative side on the graph.
(iv) To find lim(x->0+) f(x), observe the behavior of f(x) as x approaches 0 from the positive side on the graph.
(v) To determine why f(x) is not differentiable at x = 0, check if there is a sharp corner, vertical tangent, or any discontinuity at x = 0. If any of these conditions are met, then f(x) is not differentiable at x = 0.
By following these steps and analyzing the graphs obtained, you should be able to answer all the questions (a) to (d) listed above.