☰ Contents · Mathematics

Tangent and normal equations; problem solving

Lessons 6–7 · 2 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
7

Problem solving (derivatives)

Textbook, Part 1: pp. 39–41
GoalSolve mixed problems on the derivative (increments, derivative values, velocity, tangents, sign of the derivative) step by step.
New words
increment of a function · funksiya orttirmasiinstantaneous velocity · oniy tezliksign of the derivative · hosilaning ishorasipoint of momentary rest · to‘xtash nuqtasi
Explanation

A useful routine for derivative problems: 1) decide what is asked (increment, derivative value, velocity, tangent, sign); 2) write the function and differentiate it with the proper rules; 3) substitute the point or solve f′(x) = 0 / f′(x) > 0; 4) state the answer in the language of the problem, with units. For motion with position s(t), v(t) = s′(t); when v = 0 the point is momentarily at rest, and if v changes sign the direction of motion reverses. For an increment compute f(x₀ + Δx) − f(x₀) directly and compare it with f′(x₀) · Δx: for small Δx they are close. To study the sign of a derivative, factorise it and use the interval method.

Worked examples
s(t) = t³ − 6t² + 9t (m). v(t) = 3t² − 12t + 9 = 3(t − 1)(t − 3). v = 0 at t = 1 s and t = 3 s. For 1 < t < 3, v < 0 — the point moves backwards; for t < 1 and t > 3 it moves forwards. So the direction changes at t = 1 and t = 3.
f(x) = x² − 3x + 1, x₀ = 2, Δx = 0.1. f(2.1) = 4.41 − 6.3 + 1 = −0.89; f(2) = −1; the increment Δf = 0.11; Δf/Δx = 1.1. The derivative is f′(2) = 2 · 2 − 3 = 1. The ratio 1.1 is close to 1 and gets closer as Δx decreases.
Class activity

“From the word problem to the derivative”: the teacher reads word problems (“speed”, “growth rate”, “tangent to the graph”, “when it stops”). The class writes each phrase on the board as “a derivative or a condition on the derivative”.

Practice
1
For f(x) = 2x² + x find the increment of the function at x₀ = 1 with Δx = 0.2.
2
Find the values of x for which the derivative of f(x) = x³ − 12x is positive.
3
A point moves by s(t) = t² − 8t + 3 (m). At what time (s) is its velocity zero?
4
In the motion above, has the point stopped for good at t = 4 s? Explain using the sign of the velocity.