Pure Mathematics · 3rd secondary · First Term

Calculus - Behaviour of the function and curve sketching — Increasing and decreasing of the functions

About this lesson

This unit teaches how to determine the increasing and decreasing intervals of a function using the first derivative test. It explains that if f'(x) > 0 on an interval, the function is increasing there, and if f'(x) < 0, it is decreasing. The text includes worked examples with polynomial, trigonometric, and logarithmic functions, showing how to find critical points by setting the derivative to zero and then testing the sign of the derivative in each interval. It also notes that the tangent to the curve makes an acute angle with the positive x-axis in increasing intervals and an obtuse angle in decreasing intervals. Students are expected to apply this method to various functions, including those involving sine, cosine, and natural logarithms, and to verify results using graphical software like GeoGebra. The unit includes exercises for practice and a proof that a function like tan x - x is increasing on a given interval.

Main topics in this lesson

  • First derivative test for monotonic functions
  • Determining increasing and decreasing intervals
  • Applications to polynomial, trigonometric, and logarithmic functions
  • Sign of derivative and tangent angles

Key terms and vocabulary

  • Increasing function
  • Decreasing function
  • First derivative test
  • Monotonic functions
  • Critical points
  • Tangent angle

Questions and answers on Calculus - Behaviour of the function and curve sketching — Increasing and decreasing of the functions - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

الدالة f(x) = tan x - x تكون على الفترة (0، π/2):

  • 1 متزايدة
  • 2 متناقصة
  • 3 ثابتة
  • 4 غير معرفة
2

عندما تكون الدالة متناقصة على فترة ما، فإن المماس للمنحنى يصنع مع الاتجاه الموجب لمحور x زاوية:

  • 1 منفرجة
  • 2 حادة
  • 3 قائمة
  • 4 معدومة
3

عندما تكون الدالة متزايدة على فترة ما، فإن المماس للمنحنى يصنع مع الاتجاه الموجب لمحور x زاوية:

  • 1 حادة
  • 2 قائمة
  • 3 منفرجة
  • 4 معدومة
4

إذا كانت f(x) = ln x، فإن f'(x) تساوي:

  • 1 1/x
  • 2 x
  • 3 ln x
  • 4 -1/x
5

إذا كانت f(x) = cos x، فإن f'(x) تساوي:

  • 1 -sin x
  • 2 sin x
  • 3 cos x
  • 4 -cos x
6

إذا كانت f(x) = sin x، فإن f'(x) تساوي:

  • 1 cos x
  • 2 -cos x
  • 3 sin x
  • 4 -sin x

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