From f to f′ Problem set B One question at a time

The graph shows the straight line y = f(x) through (0, −3) and (3, 3).

1. (a) Write down f′(x).

   • Gradient = (3 − (−3)) ÷ (3 − 0) = 2
   • f′(x) = 2

2. (b) Does the graph of f′ cross the x-axis? Give a reason.

   • The graph of f′ is a horizontal line at y = 2
   • It is never zero, so it does not cross the x-axis

The graph shows y = f(x), a parabola with its vertex at (−1, −3). The graphs P, Q and R are shown with it. One of them is the graph of f′.

1. (a) Which of P, Q, R is the graph of f′?

   • f′ is a straight line, and f falls then rises, so f′ is negative then positive: sloping up
   • It is R

2. (b) State the x-value where the graph of f′ meets the x-axis.

   • f′ = 0 at the vertex, x = −1

Here is y = f(x), a cubic. Its turning points are marked, and three more graphs P, Q and R are drawn. One of P, Q, R is the graph of f′.

1. (a) Which of P, Q, R is the graph of f′?

   • The turning points are at x = 0 and 2, so f′ meets the x-axis there
   • f falls, rises, falls, so f′ is negative, positive, negative: an upside-down U
   • It is P

2. (b) Write down the values of x where f′(x) = 0.

   • f′ = 0 at the turning points of f
   • x = 0 and x = 2

Here is y = f(x), an M-shaped quartic. Its three turning points are marked.

1. (a) Write down the values of x where the graph of f′ meets the x-axis.

   • The graph of f′ meets the x-axis at the turning points
   • x = −2, x = 0, x = 1

2. (b) Write down the values of x for which f′(x) > 0.

   • f′ is positive where f rises
   • f′ > 0 for \(x<-2\) or \(0<x<1\)

Here is y = f(x) over one full turn (0 to 360 degrees). Its highest and lowest points are marked.

1. (a) Write down the values of x where f′(x) = 0.

   • f′ = 0 at the turning points
   • x = 0, x = 180 and x = 360

2. (b) Write down the values of x for which f′(x) > 0.

   • f falls from 0 to 180 and rises from 180 to 360: f′ is positive for 180 < x < 360

The table shows g(x) = xᵇ and its gradient function for b = 2 to 5. Use the pattern to write g′(x) for b = 6.

• The old power comes to the front, and the power drops by 1
• x⁶ → 6x⁵

The graph shows the profit P thousand pounds of a company when it makes x hundred items, for 0 ≤ x ≤ 8. The graphs J, K and L are shown with it. One of them shows the rate of change of the profit.

1. (a) Which of the three graphs shows the rate of change?

   • The rate of change is the gradient, which is f′
   • The graph rises then falls, so the gradient is positive then negative: J

2. (b) State the value of x where the rate of change is zero and say what it means.

   • The gradient is zero at the turning point, x = 4
   • The profit is at its greatest: it is neither rising nor falling there

The graph of y = f(x) is a cubic with a local maximum at (0, 0) and a local minimum at (2, −4). It rises, falls, then rises. Describe the graph of f′: its shape, where it meets the x-axis and where it is below the axis.

• Turning points of f are x-intercepts of f′: x = 0 and x = 2
• f falls between them, so f′ < 0 for 0 < x < 2
• f rises outside, so f′ > 0 there
• Positive, negative, positive is a U: an upward parabola

The graph of y = f(x) for 0 ≤ x ≤ 360 is a wave with a maximum at (90, 1) and a minimum at (270, −3). Describe the graph of f′: where it is zero, where it is positive and where it has its largest and smallest values.

• f′ = 0 at the turning points: x = 90, 270
• f rises on 0 to 90 and 270 to 360: f′ > 0; it falls on 90 to 270: f′ < 0
• f is steepest at x = 0, 180 and 360
• So f′ has a maximum at 0 and 360 and a minimum at 180