(a) Sketch, by hand, the graph of the function f(x)=e^x....

(a) Sketch, by hand, the graph of the function $f(x)=e^{x}$. paying particular attention to how the graph crosses the $y$ -axis. What fact allows you to do this? (b) What types of functions are $f(x)=e^{x}$ and $g(x)=x^{2} ?$ Compare the differentiation formulas for $f$ and $g$. (c) Which of the two functions in part (b) grows more rapidly when $x$ is large?

(a) Sketch, by hand, the graph of the function $f(x)=e^{x}$. paying particular attention to how the graph crosses the $y$ -axis. What fact allows you to do this?
(b) What types of functions are $f(x)=e^{x}$ and $g(x)=x^{2} ?$ Compare the differentiation formulas for $f$ and $g$.
(c) Which of the two functions in part (b) grows more rapidly when $x$ is large?

Key Concepts

Differentiation of Functions

Differentiation refers to the process of determining the rate at which a function changes. Exponential functions like f(x) = e^x differentiate to themselves, meaning d/dx(e^x) = e^x, which is a unique property compared to other functions, such as quadratic functions where the derivative is d/dx(x^2) = 2x. This comparison highlights the distinct behaviors in the slopes of the graphs.

Growth Rate Comparison

Growth rate comparison is a method for understanding how fast one function increases relative to another as x becomes large. Exponential functions are known to grow much more rapidly than polynomial functions, such as quadratic functions, meaning that for large values of x, f(x) = e^x will eventually surpass g(x) = x^2 by a significant margin.

Exponential Functions

Exponential functions have the form f(x) = a^x, where a > 0. A key property is that finding the value at x = 0 gives f(0) = a^0 = 1, ensuring the graph crosses the y-axis at (0,1). They are noted for their rapid growth or decay, and in particular, the function f(x) = e^x is unique because its rate of change is equal to its current value.

Graphing Techniques and y-Axis Intercepts

Graphing techniques involve identifying key points such as intercepts, asymptotes, and general behavior at the limits. For exponential functions like f(x) = e^x, knowing the y-intercept at (0,1) is crucial, as it provides a starting point for sketching the curve accurately by hand.

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