Calculus AB · Analytical Applications of Differentiation · 14 min read · Updated 2026-05-10
Sketching graphs of f, f', f'' — AP Calculus AB
AP Calculus AB · Analytical Applications of Differentiation · 14 min read
1. Relating Key Features of f and f'★★☆☆☆⏱ 4 min
The first derivative $f'$ gives the slope of the tangent line to $f$ at any point $x$, so every key behavior of $f$ translates directly to a graphical feature of $f'$, and vice versa. This topic tests your conceptual understanding of what derivatives measure, rather than just computation.
When $f$ is increasing on an interval, all tangent slopes are positive, so $f'(x) > 0$ and the graph of $f'$ lies above the $x$-axis.
When $f$ is decreasing on an interval, all tangent slopes are negative, so $f'(x) < 0$ and the graph of $f'$ lies below the $x$-axis.
Local maxima/minima of $f$ (for differentiable $f$) occur where $f'$ crosses the $x$-axis: a local maximum of $f$ means $f'$ changes from positive to negative, and a local minimum means $f'$ changes from negative to positive.
For polynomials, differentiation reduces degree by 1: a degree $n$ $f$ has a degree $n-1$ $f'$, a quick check for multiple-choice matching.
2. Relating Features of f and f' to f''★★★☆☆⏱ 4 min
The second derivative $f''$ is the derivative of $f'$, so the same $f$-$f'$ relationship applies between $f'$ and $f''$. In addition, $f''$ encodes the concavity of the original function $f$, a key graphical property.
When $f$ is concave up, it bends upward, the slope of $f$ is increasing, so $f''(x) > 0$.
When $f$ is concave down, it bends downward, the slope of $f$ is decreasing, so $f''(x) < 0$.
An inflection point occurs where $f$ changes concavity, which means $f''$ changes sign at that point. For twice-differentiable $f$, this also means $f'$ has a local extremum at that $x$-coordinate.
3. Sketching f From a Given Graph of f'★★★☆☆⏱ 4 min
A very common AP Calculus AB free-response question gives you the graph of $f'$ (often piecewise linear for easy analysis) and an initial condition $f(a) = k$, then asks you to sketch $f$ or identify its key features. Follow this step-by-step process:
Split the $x$-axis into intervals separated by $x$-intercepts of $f'$ (these are critical points of $f$).
Find the sign of $f'$ on each interval to classify critical points as local maximum, minimum, or neither.
Split intervals further by local extrema of $f'$ (these are inflection points of $f$), then find concavity of $f$ on each interval.
Calculate $y$-values of key points of $f$ using the Fundamental Theorem of Calculus: $f(b) = f(a) + \int_a^b f'(x) dx$, which simplifies to adding areas of geometric shapes when $f'$ is piecewise linear.
4. AP-Style Concept Check★★★☆☆⏱ 2 min
Common Pitfalls
Why: Students confuse critical points of f (which are x-intercepts of f') with inflection points of f (which are x-intercepts of f'' or local extrema of f').
Why: Inflection points require a sign change of f'', not just a zero value. For example, $f(x) = x^4$ has $f''(0) = 0$ but no sign change, so no inflection point.
Why: Time pressure on the exam leads to flipped relationships when working backwards.
Why: This comes from confusing critical points and inflection points under time pressure.
Why: Students forget that differentiation lowers the degree of a polynomial by 1.