Sketching f and f' from graphs Problem set B Answers

A straight line with positive gradient, crossing the x-axis at x = −1 and the y-axis at 2

A straight line with negative gradient through (−2, 0), crossing the y-axis at −4

A downward parabola crossing the x-axis at x = −1.36 and x = 2.69, and the y-axis at 11

An upward parabola crossing the x-axis at x = 0 and x = 2, and the y-axis at the origin

A curve entirely below the x-axis, increasing towards the x-axis (its asymptote), with y-intercept −4

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

Two curves, one above the other: rising to a maximum at t = 1, falling to a minimum at t = 5, then rising

1. (a) A cosine wave: starts at 2, crosses the axis at π/2 and 3π/2, minimum −2 at π, ends at 2
2. (b) An upside-down cosine wave, minimum at x = 0 and x = 2π, maximum at x = π: f(x) = −2cos 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 (−2.45, 0), (0, 0) and (2.45, 0), touching the axis at the origin, with minima at x = −1.73 and 1.73
2. (b) x < −2.45 or x > 2.45
3. (c) x < −1.73 or 0 < x < 1.73

P and Q (Q = P'), R and S (S = R')