Graph Sketching and Integrals
2
1. Sketch 𝑓(𝑥) = (𝑥 − 𝑅(𝑥 )) , where R(x) is x rounded up or down in the usual way.
Sketch 𝑔(𝑥) = 𝑓(1/𝑥)
2. 𝑥 2 + 𝑝𝑥 + 𝑝 = 0. What is the range of 𝑝 that gives real values of 𝑥? Sketch 𝑥 against 𝑝.
3. Sketch (a) 𝑦 + |𝑥 | = 1, (b) |𝑦| + 𝑥 = 1, (c) |𝑥 | + |𝑦| = 1.
4. Sketch 𝑦 = 𝑥 cos 𝑥.
𝜋
Find ∫−𝜋 𝑥 cos 𝑥 𝑑𝑥 (a) by calculation, (b) from the graph.
1
5. (a) Sketch 𝑦 = sin (𝑥) , 𝑦 2 = sin(𝑥) and 𝑦 = sin(𝑥 2 )
1
𝑑𝑦
(b) Sketch 𝑦 = 𝑥 sin (𝑥) : what happens as 𝑥 → ∞? Find 𝑑𝑥 for 𝑥 ≠ 0.
1
𝑠
−𝑠
6. By considering the graph of the 𝑓 (𝑥 ) = 𝑥 −𝑠 , show that 𝑠−1 ≤ ∑∞
≤ 𝑠−1 for 𝑠 > 1
𝑛=1 𝑛
7. Sketch (a) 𝑥 2 + 𝑦 2 = 1 and 𝑥 4 + 𝑦 4 = 1, (b) 𝑥 + 𝑦 = 1 and 𝑥 3 + 𝑦 3 = 1.
What do 𝑥 2𝑛 + 𝑦 2𝑛 = 1 and 𝑥 2𝑛+1 + 𝑦 2𝑛+1 = 1 look like for large n ?
𝑎
2
8. Sketch the function 𝐼(𝑎) defined as 𝐼 (𝑎) = ∫−𝑎 4 − 2𝑥 𝑑𝑥 for 𝑎 ≥ 0. Find all turning
points.
9. Let 𝑓(𝑥 ) = 𝑥 𝑝−1 where 𝑝 > 1.
𝑎
Show that ∫0 𝑓(𝑥) 𝑑𝑥 =
𝑎𝑝
𝑝
𝑏
and ∫0 𝑓 −1 (𝑥)𝑑𝑥 =
𝑏𝑞
𝑞
By considering a suitable graph, deduce that 𝑎𝑏 ≤
10. Sketch the graph 𝑦 =
ln 𝑥
𝑥
1
1
𝑝
𝑏𝑞
𝑞
where + = 1.
𝑎𝑝
𝑝
+
𝑞
for 𝑎, 𝑏 > 0.
and find the coordinates of its stationary point.
(a) Use your graph to find which is bigger, 𝑒 𝜋 or 𝜋 𝑒 .
(b) Use your graph to explain why the only whole number solutions to 𝑦 𝑥 = 𝑥 𝑦 with
𝑥 ≠ 𝑦 are (2, 4) and (4, 2).
4 ln 𝑥
(c) Find ∫2
𝑥
𝑑𝑥.