M171 Calculus I · Unit 1: Understanding the derivative · on Exam 1
§1.6 The second derivative
Study notes for The second derivative in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- $f''$ is the derivative of $f'$: it measures how the rate of change is changing. In motion, $s'$ is velocity and $s''$ is acceleration.
- $f'>0$: $f$ increasing. $f'<0$: $f$ decreasing. $f''>0$: $f'$ increasing, so $f$ is concave up (bends like $\cup$). $f''<0$: $f$ is concave down ($\cap$). An inflection point is where concavity changes.
- Language: "increasing at an increasing rate" means $f'>0$ and $f''>0$; "decreasing at a decreasing rate" (decreasing more slowly) means $f'<0$ and $f''>0$.
- From a graph of $f'$: $f$ is increasing where $f'$ is above the axis; $f$ is concave up where $f'$ is rising.
Key formulas
Second derivative
$$f''(x) = \frac{d}{dx}\big[f'(x)\big] = \frac{d^2y}{dx^2}$$
Concavity
$$f''>0 \Rightarrow \text{concave up},\qquad f''<0 \Rightarrow \text{concave down}$$
Worked example
A car's value $V(m)$ in dollars after $m$ miles satisfies $V(25000)=23525$, $V'(25000)=-0.61$, $V''(25000)=0.017$. Explain in everyday language.
Show the solution
After 25,000 miles the car is worth \$23,525. At that moment its value is dropping about 61 cents for each additional mile. Because $V''>0$, the rate of loss ($-0.61$ dollars/mile) is itself increasing (getting less negative): the car is still losing value, but more slowly with each mile, at a rate of $0.017$ dollars per mile per mile.
Common mistakes
- Treating $f''>0$ as "$f$ is increasing." It says $f'$ is increasing.
- Calling a point an inflection point when $f''=0$ but concavity does not change (e.g. $x^4$ at $0$).
- Mixing up "decreasing at an increasing rate" (dropping faster) with "decreasing at a decreasing rate" (levelling off).
Exam tipFill a sign chart with three rows ($f$, $f'$, $f''$) before writing sentences. Exam 1 practice problem 22 is exactly this.
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