Sketching f and f' from graphs Problem set A Answers

A straight line with negative gradient, crossing the x-axis at x = 2 and the y-axis at 4

A straight line with positive gradient through (3, 0), crossing the y-axis at −6

An upward parabola crossing the x-axis at x = −1.12 and x = 1.79, and the y-axis at −6

An upward parabola crossing the x-axis at x = −1 and x = 1, and the y-axis at −3

A curve entirely above the x-axis, decreasing towards the x-axis (its asymptote), with y-intercept 3

1. (a) An increasing curve above the x-axis that gets steeper, such as \(f(x)=e^x\)
2. (b) An increasing curve that gets less steep, such as \(f(x)=-e^{-x}\) (levelling off along y = 0 from below)

Two increasing S-shaped curves, one above the other, flat at t = 0 and t = 6, steepest at t = 3

1. (a) A cosine wave: starts at 3, crosses the axis at π/2 and 3π/2, minimum −3 at π, ends at 3
2. (b) An upside-down cosine wave, minimum at x = 0 and x = 2π, maximum at x = π: f(x) = −3cos x + c
3. (c) Adding any constant c moves the graph up or down but does not change its gradient, so every value of c gives the same f'

1. (a) A W-shape through (−2, 0), (0, 0) and (2, 0), touching the axis at the origin, with minima at x = −1.41 and 1.41
2. (b) x < −2 or x > 2
3. (c) x < −1.41 or 0 < x < 1.41

A and B (B = f'), C and D (D = f')