Applications of Derivatives

A critical point of $f$ occurs at $x=c$ where $f'(c) = 0$ or $f'(c)$ is undefined (and $f(c)$ exists).

The first derivative test: examine the sign of $f'(x)$ on either side of the critical point.

If $f'$ changes from $+$ to $-$ at $c$, then $f(c)$ is a local maximum.

If $f'$ changes from $-$ to $+$ at $c$, then $f(c)$ is a local minimum.

If $f'$ does not change sign (e.g., $f(x) = x^3$ at $x=0$), the critical point is neither a max nor a min.

To apply: find all critical points, then build a sign chart for $f'(x)$ using test values in each interval.

An alternative to the first derivative test that avoids building a sign chart. At a critical point where $f'(c) = 0$:

If $f''(c) > 0$: the curve is concave up (bowl-shaped) at $c$, so $f(c)$ is a local minimum. The function is curving upward — any nearby point is higher.

If $f''(c) < 0$: the curve is concave down (arch-shaped) at $c$, so $f(c)$ is a local maximum. The function is curving downward — any nearby point is lower.

If $f''(c) = 0$: the test is inconclusive. The critical point could be a min, max, or neither. Fall back to the first derivative test. Example: $f(x) = x^4$ at $x=0$ has $f''(0)=0$ but it's still a minimum. Meanwhile $f(x)=x^3$ at $x=0$ has $f''(0)=0$ and it's neither.

When to use which: the second derivative test is faster when $f''$ is easy to compute. The first derivative test (sign chart) is more reliable — it never fails and also works when $f'$ is undefined.

Concavity describes how the curve bends. $f''(x) > 0$ means concave up (opening upward), $f''(x) < 0$ means concave down (opening downward).

An inflection point is where the concavity changes: $f''$ switches sign. At an inflection point, the curve transitions from bending one way to bending the other.

To find inflection points: set $f''(x) = 0$ (or find where $f''$ is undefined), then verify $f''$ changes sign across that point.

Example: $f(x) = x^3$ has $f''(x) = 6x$. This equals $0$ at $x=0$, and changes from negative to positive. So $(0, 0)$ is an inflection point.

Note: $f''(c) = 0$ alone is not enough. $f(x) = x^4$ has $f''(0) = 0$ but no inflection point (concavity doesn't change).

The Extreme Value Theorem (EVT): if $f$ is continuous on a closed interval $[a,b]$, then $f$ attains an absolute maximum and an absolute minimum somewhere on $[a,b]$. Both 'continuous' and 'closed interval' are required — remove either condition and the theorem can fail.

The closed interval method to find them:

Step 1: Find all critical points of $f$ in the open interval $(a,b)$ — where $f'(x)=0$ or $f'$ is undefined.

Step 2: Evaluate $f$ at each critical point and at both endpoints $a$ and $b$. Build a table of $(x, f(x))$ values.

Step 3: The largest $f$-value in the table is the absolute maximum, the smallest is the absolute minimum.

This is the go-to method for any 'find the max/min on an interval' problem. The absolute extrema must occur at a critical point or an endpoint — there's nowhere else for them to hide. Never forget the endpoints!

A complete curve sketch synthesizes everything you've learned about derivatives into a picture. Follow this checklist systematically:

1. Domain and intercepts: where is $f$ defined? Set $f(x)=0$ for $x$-intercepts, evaluate $f(0)$ for the $y$-intercept.

2. Symmetry: is $f$ even ($f(-x)=f(x)$, symmetric about $y$-axis), odd ($f(-x)=-f(x)$, symmetric about origin), or neither? This can halve your work.

3. First derivative analysis: find critical points (where $f'=0$ or undefined). Build a sign chart to determine intervals of increase ($f'>0$) and decrease ($f'<0$). Classify each critical point as local max, local min, or neither.

4. Second derivative analysis: find inflection point candidates (where $f''=0$ or undefined). Build a sign chart for $f''$ to determine concavity (up where $f''>0$, down where $f''<0$). Verify sign changes for actual inflection points.

5. Asymptotes: vertical (denominator → 0 with nonzero numerator), horizontal ($\lim_{x\to\pm\infty}$), and oblique/slant (when degree of numerator = degree of denominator + 1).

6. Plot key points (intercepts, critical points, inflection points) and connect the dots, respecting the increase/decrease direction and concavity at every point.

The general strategy for optimization word problems:

Step 1: Draw a diagram and define variables. Label everything.

Step 2: Write the objective function (what you want to maximize or minimize).

Step 3: Write the constraint equation (the relationship that limits your variables).

Step 4: Use the constraint to eliminate one variable from the objective, getting a function of a single variable.

Step 5: Find the critical points of this function. Test them (and the endpoints, if the domain is closed) to identify the optimum.

Step 6: Answer the original question. Make sure you're solving for what was asked.

Common setups: maximize area given a perimeter constraint, minimize material given a volume constraint, maximize revenue.

Related rates problems involve multiple quantities changing over time, connected by a geometric or physical equation.

Step 1: Draw a picture and identify all variables. Label which quantities are changing.

Step 2: Write an equation relating the variables (e.g., Pythagorean theorem, area formula, volume formula).

Step 3: Differentiate both sides with respect to time $t$ using implicit differentiation.

Step 4: Plug in all known values and rates at the specific instant in question.

Step 5: Solve for the unknown rate.

Critical: never plug in specific values before differentiating. The equation must remain general during differentiation because the variables are changing.

Common setups: expanding balloon (sphere volume), sliding ladder (Pythagorean theorem), filling cone (similar triangles + cone volume), spreading oil slick (circle area).

Given a position function $s(t)$: velocity is $v(t) = s'(t)$ and acceleration is $a(t) = v'(t) = s''(t)$. This is the most natural physical application of derivatives.

The particle is moving right (positive direction) when $v(t) > 0$ and moving left when $v(t) < 0$. The particle is at rest (momentarily stopped) when $v(t) = 0$.

Direction change: the particle changes direction when $v(t) = 0$ and the velocity changes sign. If $v$ doesn't change sign, the particle merely pauses.

Speed vs. velocity: velocity is signed (includes direction), speed is $|v(t)|$ (always non-negative). The particle speeds up when velocity and acceleration have the same sign (both positive or both negative) and slows down when they have opposite signs.

Displacement vs. total distance: $\int_a^b v(t)\,dt$ gives displacement (net change in position, which can be zero if the particle returns). $\int_a^b |v(t)|\,dt$ gives total distance traveled (always ≥ 0). To compute total distance, find where $v(t) = 0$, split the integral, and flip signs on intervals where $v < 0$.

The linearization of $f$ at $a$ is $L(x) = f(a) + f'(a)(x-a)$. This is the equation of the tangent line at $x = a$, repurposed as an approximation tool.

For values of $x$ near $a$, $f(x) \approx L(x)$. The closer $x$ is to $a$, the better the approximation. This is the foundation of how engineers and scientists do quick estimates without a calculator.

Differentials formalize this: if $y = f(x)$, then $dy = f'(x)\,dx$ is the approximate change in $y$ when $x$ changes by a small amount $dx$. The actual change is $\Delta y = f(x+dx) - f(x)$, and $dy \approx \Delta y$ when $dx$ is small.

Why this works: the tangent line is the best linear approximation to the curve at that point. For tiny changes, a curve behaves almost exactly like its tangent line. The error is proportional to $(x-a)^2$, so it vanishes quickly as $x \to a$.

Applications: estimating values like $\sqrt{4.02}$ or $\sin(0.1)$ by hand, error propagation in measurements, and building the intuition that leads to Taylor series (which extend this idea to quadratic, cubic, and higher-degree approximations).

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c$ in $(a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

In words: somewhere between $a$ and $b$, the instantaneous rate of change equals the average rate of change.

Geometric interpretation: there's a point where the tangent line is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$.

Applications: proving that a function must have a certain derivative value, establishing speed limits (if you drove 120 miles in 2 hours, you must have been going 60 mph at some instant).

Newton's method finds approximate roots of $f(x) = 0$ using tangent lines.

Start with an initial guess $x_0$. The iteration formula is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.

Each step draws the tangent line at $(x_n, f(x_n))$ and finds where it crosses the $x$-axis. That crossing point is $x_{n+1}$.

Convergence is typically quadratic: the number of correct digits roughly doubles each step. For example, if $x_3$ has 3 correct digits, $x_4$ may have 6.

Error estimation: $|x_{n+1} - r| \approx \frac{|f''(r)|}{2|f'(r)|} \cdot |x_n - r|^2$ where $r$ is the true root. This quantifies the quadratic convergence.

Pitfalls: the method can fail if $f'(x_n) = 0$ (horizontal tangent), if the initial guess is too far from the root, or if the function oscillates. Cycling is possible — Newton's method applied to $x^3 - 2x + 2$ starting at $x_0 = 0$ enters an infinite loop between $1$ and $0$.

Stopping criterion: iterate until $|x_{n+1} - x_n| < \varepsilon$ for a desired tolerance $\varepsilon$, or until $|f(x_n)| < \varepsilon$.

The derivative is the universal language for rates. Here are key applications across disciplines.

Physics — velocity and acceleration: if $s(t)$ is position, then $v(t) = s'(t)$ is velocity and $a(t) = v'(t) = s''(t)$ is acceleration. Force equals mass times acceleration: $F = ma = ms''(t)$.

Biology — population growth: if $P(t)$ is population, $P'(t)$ is the growth rate. The per-capita growth rate is $P'(t)/P(t)$. This leads to the logistic equation $P' = rP(1 - P/K)$.

Chemistry — reaction rates: the rate of a chemical reaction $\frac{d[A]}{dt}$ measures how fast a reactant $A$ is consumed. The rate law relates this to concentrations: $-\frac{d[A]}{dt} = k[A]^n$.

Economics — marginal analysis: the marginal cost $C'(x)$ is the cost of producing one additional unit. The marginal revenue $R'(x)$ is the revenue from one more sale. Profit is maximized when $R'(x) = C'(x)$.

Engineering — sensitivity analysis: if an output $y$ depends on a parameter $p$, the derivative $dy/dp$ measures how sensitive the output is to small changes in $p$. This is fundamental to error propagation: $\Delta y \approx |y'(p)| \cdot \Delta p$.

The mathematical tools (chain rule, implicit differentiation, related rates) are identical across all these fields. Only the variable names change.