Curve Sketching and Optimization Problems Math1241 Calculus I

Curve Sketching and Optimization Problems
MAT1241: Calculus 1
University of Eswatini.


Curve Sketching:
Curve sketching involves analyzing the behavior of a function and drawing its graph. Here are the key
steps:

1. Domain and Symmetry:
○ Determine the domain of the function.
○ Check for symmetry (even, odd, or neither).
2. Intercepts:
○ Find the (x)-intercepts (set (f(x)) equal to zero and solve for (x)).
○ Find the (y)-intercept (evaluate (f(0))).
3. Asymptotes:
○ Vertical asymptotes occur where the denominator is zero (if any).
○ Horizontal asymptotes depend on the degree of the numerator and denominator.
4. Critical Points and Inflection Points:
○ Find critical points by solving (f’(x) = 0).
○ Determine concavity using the second derivative test.
5. Sketch the Graph:
○ Use the information above to draw the graph.

Example 1:

Consider the function .

1. Domain and Symmetry:
○ Domain: All real numbers.
○ Neither even nor odd.
2. Intercepts:
○ (x)-intercepts: Set (f(x) = 0): (x = 0).
○ (y)-intercept: (f(0) = 0).
3. Asymptotes:
○ Vertical asymptote: None.
○ Horizontal asymptote: As , .
4. Critical Points and Inflection Points:
○ Find .
○ Solve (f’(x) = 0): No critical points.

○ Second derivative: .
○ Concavity: (f’‘(x) > 0) for (x < -1) and (f’'(x) < 0) for (-1 < x < 1).