Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/differentiation/12-9-1/problems/b/

jmaths

IB Mathematics: Applications and Interpretation HL · 12.9: Differentiation · 12.9.1

Name Date

Increasing and decreasing Problem set B

Read the turning points from the graph first. Give intervals as inequalities in x.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The graph shows y = f(x), a parabola with its lowest point at (−2, −3). Write the values of x for which f is increasing.

    −5−4−3−2−11−4−2246xyy = f(x)(−2, −3)

  2. The graph shows y = f(x), a cubic with turning points at x = 0 and x = 3. Write down the values of x for which f is:

    −2−112345−101020xyy = f(x)(0, 0)(3, 13.5)
    1. (a) increasing;
    2. (b) decreasing.

  3. The graph shows y = f(x) for 0 ≤ x ≤ 360, where x is in degrees. It has maxima at (0, 2) and (360, 2) and a minimum at (180, −4).

    90180270360−4−22x (degrees)yy = f(x)(0, 2)(180, −4)(360, 2)
    1. (a) Write down the values of x for which f is increasing.
    2. (b) Write down the values of x for which the gradient of the graph is zero.

  4. The line shown is the tangent to the curve at P(1, 2). It passes through (0, 4) and (2, 0).

    −2−1123456−22468xyy = f(x)P
    1. (a) Find the gradient of the curve at P.
    2. (b) Write down the value of x at which the gradient of the curve is zero.

In context

  1. The graph shows the depth D centimetres of water in a tank t hours after 8 am, for 0 ≤ t ≤ 8. The turning points are (2, 8) and (6, 40).

    1234567810203040t (hours)D (cm)D(2, 8)(6, 40)
    1. (a) For which values of t is the depth of water decreasing?
    2. (b) State the gradient of the graph at t = 6 and say what it tells you about the water.

Stretch

  1. A function f is defined for all x. It is decreasing for x < 0, increasing for 0 < x < 4 and decreasing for x > 4. Describe the graph of y = f(x): where the gradient is zero, what kind of turning points they are, and the overall shape.

  2. The graph shows y = f(x). It is a quartic with two minima and one maximum.

    −4−3−2−11234246810xyy = f(x)(−3, 0)(0, 9)(3, 0)
    1. (a) Write down the values of x for which f is increasing.
    2. (b) Write down the values of x at which the gradient is zero.

jmaths.xyz · AI HL · 12.9.1 Gradient, increasing and decreasing