Sketch the graph of the function using the curve-sketching guidelines.

Question

Sketch the graph of the function  using the curve-sketching guidelines.

Quizplus · Accepted Answer

We can use the following steps to sketch the graph of the function:

1. Determine the domain and range of the function:
The function is an exponential function, which means that it is defined for all real numbers. The range of the function is (0, infinity), since e raised to any power is always positive.

2. Find the x- and y-intercepts:
The x-intercept occurs when y = 0, which is not possible since e raised to any power is always positive. Therefore, there are no x-intercepts. The y-intercept occurs when x = 0, which gives f(0) = 11e^0 = 11, so the y-intercept is (0, 11).

3. Find the critical points:
A critical point is a point where the first derivative of the function is zero or undefined. Since the first derivative of an exponential function is always positive, there are no critical points.

4. Find the intervals of increase and decrease:
Since the first derivative is always positive, the function is increasing on its entire domain.

5. Find the intervals of concavity and points of inflection:
The second derivative of the function is f''(x) = 11e^x, which is always positive. Therefore, the function is concave up on its entire domain and there are no inflection points.

6. Sketch the graph:
Based on the above analysis, we know that the function is an increasing exponential function that is concave up. The y-intercept is (0, 11). Therefore, the graph must look something like this:

graph{11e^x [-10, 10, -5, 15]}

Sketch the Graph of the Function Using the Curve-Sketching