Curves of f , f’ , f” and Curve Sketching

To sketch the graph of a function effectively using calculus, you typically follow these steps:

1. **Analyze the function:** Determine the domain, range (if possible), intercepts (x and y), and check for symmetry (even/odd function) and periodicity.
2. **Find the first derivative (f'(x)):** Use f'(x) to find critical points (where f'(x) = 0 or is undefined) and intervals where the function is increasing (f'(x) > 0) or decreasing (f'(x) < 0). Use the First Derivative Test to classify local extrema.
3. **Find the second derivative (f''(x)):** Use f''(x) to find possible inflection points (where f''(x) = 0 or is undefined) and intervals where the function is concave up (f''(x) > 0) or concave down (f''(x) < 0). Confirm inflection points where concavity changes.
4. **Find asymptotes:** Identify vertical asymptotes (using infinite limits) and horizontal asymptotes (using limits at infinity). Check for slant asymptotes if applicable.
5. **Plot key points:** Plot intercepts, critical points, inflection points, and points related to asymptotes.
6. **Sketch the curve:** Connect the points, following the behavior determined from the analysis of the derivatives and asymptotes.

• **First Derivative (f'(x)):** Tells you about the function's slope. Where f'(x) is positive, the function is going up (increasing). Where f'(x) is negative, it's going down (decreasing). Critical points (where f'(x)=0 or is undefined) are candidates for local maximums or minimums, which can be confirmed using the First Derivative Test (sign change of f'(x)) or the Second Derivative Test.
• **Second Derivative (f''(x)):** Tells you about the function's concavity. Where f''(x) is positive, the function's graph is concave up (like a smile). Where f''(x) is negative, it's concave down (like a frown). Points where concavity changes are called inflection points (provided the function is defined there).

These derivatives provide the shape and turning points of the graph.

• **Intercepts:** These are points where the graph crosses the x-axis (x-intercepts, where y=0) or the y-axis (y-intercept, where x=0). They provide specific points the graph must pass through.
• **Asymptotes:** These lines describe the behavior of the graph as it approaches boundaries or extends infinitely. Vertical asymptotes indicate where the function value shoots up or down to infinity. Horizontal or slant asymptotes describe the end behavior of the graph as x goes to positive or negative infinity. Asymptotes act as guides that the curve approaches.

Knowing the domain tells you where the function is defined and where you can expect to find the graph. You should only sketch the graph within its domain. Points outside the domain might indicate vertical asymptotes, holes, or simply regions where the function does not exist. Understanding the domain is the essential first step to avoid attempting to sketch where the function isn't defined.