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AP Calculus · Unit 5 · Analytical applications of differentiation · last updated

Analytical applications of differentiation practice questions (AP Calculus)

Unit 5 covers the Mean Value Theorem and Extreme Value Theorem, critical points, the first and second derivative tests, concavity and points of inflection, curve sketching, and optimization. Below is the one free AP Calculus question on it from our free exams, with its worked answer behind a toggle, then 6 key terms with the formulas typeset, and a door to make an AP Calculus-style paper on this unit from your own notes. The free exams are typeset PDFs with an answer key; the exam downloads without a signup.

Make a practice exam from your notes →Upload your notesAP Calculus style preselected; tick "Unit 5 · Analytical applications of differentiation" on the next screen. Free to try, no card.

Sample questions

The one free AP Calculus question on this unit, from the free AP Calculus AB Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

  1. Question 12 marks · Mean Value Theorem
    For f(x)=x2f(x) = x^2 on the interval [1,3][1, 3], the value of cc guaranteed by the Mean Value Theorem satisfies f′(c)=f'(c) = the average rate of change. Then c=c =
    • A.
      32\frac{3}{2}
    • B.
      22
    • C.
      5\sqrt{5}
    • D.
      52\frac{5}{2}
    From the free AP Calculus AB Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for analytical applications of differentiation, typeset as they appear on the paper.

Mean Value Theorem (MVT)
If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), some cc in (a,b)(a, b) has instantaneous rate equal to the average rate. Both hypotheses must be stated to earn the point.
f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
Extreme Value Theorem (EVT)
A function continuous on a closed interval attains an absolute maximum and an absolute minimum there. The candidates are the critical points inside the interval and the two endpoints.
Critical point
A point in the domain where f′(x)=0f'(x) = 0 or f′(x)f'(x) does not exist. Every local extremum of a differentiable function is at a critical point, but not every critical point is an extremum.
First derivative test
At a critical point, if f′f' changes from positive to negative there is a local maximum; from negative to positive, a local minimum. The justification sentence must mention the sign change.
Second derivative test and concavity
If f′(c)=0f'(c) = 0 and f′′(c)>0f''(c) > 0 there is a local minimum; if f′′(c)<0f''(c) < 0, a local maximum. The graph is concave up where f′′>0f'' > 0, concave down where f′′<0f'' < 0, with a point of inflection where f′′f'' changes sign.
f′′(x)>0⇒concave up,f′′(x)<0⇒concave downf''(x) > 0 \Rightarrow \text{concave up}, \qquad f''(x) < 0 \Rightarrow \text{concave down}
Optimization
Express the quantity to optimise as a function of one variable using a constraint, find the critical points, and confirm the extremum with a derivative test or the candidates test. State the answer in the units asked for.

Frequently asked questions

Are there free analytical applications of differentiation practice questions for AP Calculus?

Yes. 1 question on analytical applications of differentiation appears in the free AP Calculus AB Practice Exam, and the sample questions on this page come from it with worked answers. Each exam can be read online or downloaded as a typeset PDF without signing up, and its full answer key is emailed on request.

How do I make an AP Calculus practice paper on analytical applications of differentiation from my notes?

Open the create page with the AP Calculus style preselected, tick "Unit 5 · Analytical applications of differentiation" under the units to cover (or upload your notes and slides on it), choose the question mix and time limit, and generate. You get a typeset PDF in the AP format with a separate answer key, in about a minute. It is free to try, no card needed.

What does AP Calculus expect you to know about analytical applications of differentiation?

Unit 5 covers the Mean Value Theorem and Extreme Value Theorem, critical points, the first and second derivative tests, concavity and points of inflection, curve sketching, and optimization. The terms to have cold: Mean Value Theorem (MVT), Extreme Value Theorem (EVT), Critical point, First derivative test, Second derivative test and concavity. Each is defined below with its formula typeset where one applies.

What is the AP Calculus exam format?

The AP Calculus guide explains the papers, their timing and mark allocation, with a sample question and mark scheme and the common mistakes, and links to the generator with the AP Calculus style preselected.

Other AP Calculus topics

All AP Calculus topics · AP Calculus exam guide · All free practice exams

Make an AP Calculus paper on analytical applications of differentiation from your notes

Open "Pick your exam" with the AP Calculus style preselected and tick "Unit 5 · Analytical applications of differentiation", or upload your notes on it. You get a typeset paper in the AP format with a separate answer key.

Make a practice exam from your notes →Upload your notes