What does derivatives tell us about the graph of a function (continuous)

Overestimation or Underestimation

Understanding the behavior of a curve allows us to determine whether a linear approximation is an overestimation or an underestimation.

Proposition. Let $L(x)$ be a linear approximation of $f(x)$ at $x=a$, and let $a’$ be a number close to $a$. If $f’’$ is positive over an interval containing both $a$ and $a’$, then $L(a’)$ is an overestimation. Conversely, if $f’’$ is negative over an interval containing both $a$ and $a’$, then $L(a’)$ is an underestimation.

Inflection Point

An inflection point of the curve $y = f(x)$ is defined as a point $(x, f(x))$ where $f’‘(x) = 0$, and $f’’$ changes sign on both sides of $x$.

Slant asymptote

A slant asymptote of a function $f(x)$ is defined by the following condition:

\[\lim_{x\to\infty} [f(x) - (ax + b)] = 0.\]

For rational functions, where $f(x) = \frac{F(x)}{G(x)}$, with both $F$ and $G$ being polynomials, a slant asymptote exists when the degree of the numerator $F$ is one greater than the degree of the denominator $G$ (i.e., $\deg(F) = \deg(G) + 1$). In such cases, $f(x)$ can be expressed as $ax + \frac{R(x)}{G(x)}$, and the slant asymptotes can be given as:

\[y = ax + \lim_{x\to\infty}\frac{R(x)}{G(x)}.\]

For non-rational functions, the existence of slant asymptotes can be determined by checking the limit:

\[\lim_{x\to\infty}\frac{f(x)}{x}\]

(or $\lim_{x\to-\infty}\frac{f(x)}{x}$). If this limit exists, it implies that there may be a slant asymptote of the form $ax + b$, where $a$ is the limit and $b=\displaystyle\lim_{x\to\infty}f(x)-ax$.

Curve Sketching in Calculus 1

Curve sketching is a valuable tool to test your comprehension of the material introduced in Calculus 1. When sketching a curve, you need to gather the following essential information:

1. Calculate $y’$ and $y’’$

| Aspect | Details | |——–|———| | Domain | 1. Find the domain of $f(x)$ | | Asymptotes | 2. Determine vertical asymptotes | | $y$-Intercept | 3. Find the $y$-intercept (0, f(0)) | | $x$-Intercepts | 4. Find the $x$-intercepts (Solve $f(x) = 0$) | | Critical Points | 5. Find critical points (Solve $y’ = 0$ and locate where $f’$ is undefined) | | Inflection Points | 6. Find potential inflection points (Solve $y’’ = 0$ and locate where $y’’$ is undefined) |

| Aspect | Details | |——–|———| | Increasing/Decreasing | 1. Identify increasing intervals ($f’ > 0$) and decreasing intervals ($f’ < 0$) | | Concavity | 2. Determine concave up intervals ($f’’ > 0$) and concave down intervals ($f’’ < 0$) | | Extreme Points | 3. Utilize the above information to identify local maximum/minimum points and inflection points |

3. Consider Behavior at Infinity

  • Determine if a horizontal or slant asymptote exists.

This structured approach will help you effectively sketch curves in calculus and understand the key aspects of a function’s behavior.

Optimization Problems

An optimization problem requires you to find the maximum or minimum value of a quantity. To tackle such problems, follow these steps:

Step 1. Building a Model: Use the information provided in the question to create a function that describes the relationship between the dependent and independent variables.

Step 2. Find Local Extremes: Determine the local maximums and minimums of the function.

Step 3. Apply Common Sense: Eliminate answers that are not feasible based on the problem’s context. Then, identify the global maximum or minimum of the function, which serves as the final solution.