☰ Contents · Mathematics

Applications of the derivative: problem solving

Lessons 12 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
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Problem solving (applications of the derivative)

Textbook, Part 1: pp. 73–78
GoalConsolidate the chapter on the derivative by solving mixed problems (investigation, tangent, velocity and acceleration, greatest/least value) with one overall plan.
New words
investigation plan · tekshirish rejasiacceleration · tezlanishgreatest value on a segment · kesmadagi eng katta qiymatcheck · tekshiruv
Explanation

In a mixed problem first translate each request into the meaning of the derivative: “increase” means f′ > 0, “extremum” means f′ = 0 with a sign change, “tangent” means slope f′(x₀), “velocity” means s′, “acceleration” means s″ or v′. One function then gives a lot of information: factorise f′ and study its sign with the interval method; that is enough for intervals of monotonicity, extrema and the greatest/least value on a segment. Check every result: a local maximum value must exceed nearby values, and a tangent’s slope must match the graph rising or falling. Give the answer with units and in the language of the problem. The greatest value on a segment may sit at an endpoint, even if the derivative is not zero there.

Worked examples
f(x) = 2x³ + 3x² − 36x + 10. f′ = 6x² + 6x − 36 = 6(x + 3)(x − 2). Increasing on (−∞, −3) and (2, ∞); decreasing on (−3, 2). Maximum f(−3) = 91, minimum f(2) = −34. On [−4, 3]: f(−4) = 74, f(−3) = 91, f(2) = −34, f(3) = −17; greatest 91, least −34. Tangent at x = 0: f(0) = 10, f′(0) = −36, so y = −36x + 10.
A stone is thrown upwards: h(t) = 20t − 5t² (m, t in seconds). v = 20 − 10t, a = −10 m/s². v = 0 at t = 2 s: the greatest height is h(2) = 40 − 20 = 20 m. It lands when h = 0 ⇒ t = 4 s, with velocity v(4) = −20 m/s (the sign means downwards).
Class activity

“Derivative map”: each group writes one central function (say x³ − 12x) on the board and draws around it everything the derivative can reveal: monotonicity, extrema, tangents, velocity, greatest value. Groups swap maps and hunt for mistakes.

Practice
1
Find the intervals where f(x) = x³ − 3x² increases.
2
A stone moves by h(t) = 30t − 5t² (m). Find the greatest height (m).
3
Write the equation of the tangent to y = √x at x₀ = 4.
4
Where is the greatest value of f(x) = x on [0, 1], and why is it not at a point where the derivative is zero?