Question

The graph of f(x) = e^x curves upward in the interval from x = -1 and x = 1. Interpreting f'(x) = e^x as the slopes of tangent lines and noting that the larger x is, the larger e^x is, explain why the graph curves upward. For larger values of x, the graph of f(x) = e^x appears to shoot straight up with no curve. Using the tangent line, determine if this is correct or just an optical illusion.

Answer 1

@Mertsj can u help?

Answer 2

f'(x) is the slope function. More precisely, its value for any given x is the slope of the tangent line to the curve at that point. If the slope is positive we know that the line is rising as we read the graph from left to right and so we know that the graph curves upward. That is not the reason the graph curves upward, it is how we know the graph curves upward without looking at the graph.

Answer 3

For larger values of x, the slope is large but unless the slope is undefined (as it is for a vertical line), the graph is not going straight up.

Answer 4

okay

Answer 5

Does it make any sense?

Answer 6

yes