Answer to Question #248929 in Calculus for Sam

Question #248929
The graph of f(x) = ex curves upward in the interval from x = - 1 to x = 1 Interpreting f(x) = ex as the slopes of tangent lines and noting that the larger x is, the larger exis, explain why the graph curves upward. For larger values of x, the graph of f(x) = ex appears to shoot straight up with no curve. Using the tangent line, determine whether this is correct or just an optical illusion.
1
Expert's answer
2021-10-12T11:06:58-0400

A graph is said to be concave up (upward) at a point if the tangent line to the graph at that point lies below the graph.

When the function y = f (x) is concave up, the graph of its derivative y = f '(x) is increasing, i.e.

the graph of y = f (x) is concave upward on those intervals where y = f "(x) > 0.

We have "f''(x)=e^x>0" , so the graph of f(x) = ex curves upward.


Tangent line for f(x):

"y-y_0=e^{x_0}(x-x_0)"

"g(x)=e^{x_0}(x-x_0)+y_0"

"x=(g(x)-y_0)\/e^{x_0}+x_0"

So, for larger values of x:

"x\\to x_0"

So, tangent line of f(x) looks like straight vertical line.

For larger values of x, the graph of f(x) = ex appears to shoot straight up with no curve, and it is not  optical illusion.


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