Lessons 1 · 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
1
Ratio of increments of variable quantities. Definition of a tangent. Increment of a function
Textbook, Part 1: pp. 3–11
GoalCompute the ratio of increments (the average rate of change), see it as the slope of a secant, and picture the tangent as the limiting position of the secant.
When the argument changes from a to a + h, the increment of the function is f(a + h) − f(a) and the increment of the argument is h. The ratio (f(a + h) − f(a))/h is called the difference quotient and gives the average rate of change of f on [a, a + h]. Geometrically it equals the slope of the secant through the graph points (a, f(a)) and (a + h, f(a + h)). For a linear function y = kx + b the ratio is the same on every interval, namely k; for a non-linear function it depends on the interval. If the second point is moved along the graph towards the first (h gets smaller), the secant approaches a limiting position, and that line is regarded as the tangent to the graph at (a, f(a)). So the slope of the tangent is the number the difference quotients approach.
Worked examples
A point moves by s(t) = 3t² + 2 (m). Its average speed on [1, 3] seconds is: s(3) = 29, s(1) = 5, (29 − 5)/(3 − 1) = 12 m/s. On [1, 1 + h]: (3(1 + h)² + 2 − 5)/h = (6h + 3h²)/h = 6 + 3h. For h = 0.1 this is 6.3; for h = 0.01 it is 6.03; as h shrinks the ratio approaches 6.
For f(x) = 2x − 5 the ratio on [0, 4] is (3 − (−5))/4 = 2 and on [10, 11] it is (17 − 15)/1 = 2: always 2. For f(x) = x³ the ratio on [0, 2] is (8 − 0)/2 = 4 and on [1, 2] it is (8 − 1)/1 = 7: it depends on the interval.
Class activity
“From secant to tangent”: draw the parabola y = x² on the board and mark A(1, 1). Each student picks a point B of their own (for example x = 3, 2, 1.5, 1.1) and computes the slope of the secant AB. Collect the results in a table and decide together which number they approach.
Practice
1
Find the increment f(a + h) − f(a) of f(x) = x² − 4x at a = 3 with h = 0.5.
Compute the ratio of increments of y = −4x + 7 on [2, 9].
-4
3
Find the average rate of change of f(x) = x² + x on [1, 4].
6
4
Why does the slope of a secant not depend on the interval for a linear function but does for a parabola?
The graph of a linear function is a straight line, so any secant through two of its points is that same line. A parabola is curved, so secants through different pairs of points have different slopes.