Question

(a) Sketch, by hand, the graph of the function \(f\left( x \right) = {e^x}\), paying particular attention to how the graph crosses the \(y\)-axis. What is the slope of the tangent line at that point?

(b) What types of functions are\(f\left( x \right) = {e^x}\)and\(g\left( x \right) = {x^e}\)? Compare the differentiation formulas for\(f\)and\(t\).

(c) Which of the two functions in part (b) grows more rapidly when \(x\) is large?

Short answer

(a) The graph of the function \(f\left( x \right) = {e^x}\) is:

(b) \(f\left( x \right) = {e^x}\) is an exponential function and \(g\left( x \right) = {x^e}\) is a power function. The differentiation of the functions are \(f'\left( x \right) = {e^x}\) and \(g'\left( x \right) = e{x^{e - 1}}\).

(c) When the value of \(x\) grows rapidly the function grows more rapidly.

Step-by-step solution

(a) Step 1: Sketch, by hand, the graph of the function \(f\left( x \right) = {e^x}\), paying particular attention to how the graph crosses the \({\bf{y}}\)-axis.

By using graphing calculator, the sketch of the function \(f\left( x \right) = {e^x}\) is shown below:

(b) Step 2: Compare the differentiation formulas for \(f\) and \(t\)

Consider the function \(f\left( x \right) = {e^x}\)which is an exponential function and the function \(g\left( x \right) = {x^e}\) which is a power function.

Now differentiate the function \(f\left( x \right) = {e^x}\)and \(g\left( x \right) = {x^e}\).

\(\begin{array}{c}\frac{{d\left( {f\left( x \right)} \right)}}{{dx}} &=& \frac{d}{{dx}}\left( {{e^x}} \right)\\ &=& {e^x}\end{array}\)

And,

\(\begin{array}{c}\frac{{d\left( {g\left( x \right)} \right)}}{{dx}} &=& \frac{d}{{dx}}\left( {{x^e}} \right)\\ &=& e{x^{e - 1}}\end{array}\)

Thus, \(f\left( x \right) = {e^x}\)is an exponential function and \(g\left( x \right) = {x^e}\)is a power function.

\(f'\left( x \right) = {e^x}\) and \(g'\left( x \right) = e{x^{e - 1}}\).

(c) Step 3: Determine which of the two functions in part (b) grows more rapidly when \({\bf{x}}\) is large

Consider the function \(f\left( x \right) = {e^x}\)and \(g\left( x \right) = {x^e}\).

As the differentiation of the functions are \(f'\left( x \right) = {e^x}\)and \(g'\left( x \right) = e{x^{e - 1}}\).

When the value of \(x\) grows rapidly the function grows more rapidly.