MAT137Y1 – LEC0501
Calculus!
The Fundamental Theorem of Calculus Part 1
January 21st, 2019
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 1
For next lecture
For Wednesday (Jan 23), watch the videos:
• FTC – 2: 8.5, 8.6, 8.7
• Integration by substitution: 9.1, (9.2), (9.3), (9.4)
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 2
True or False?
Let𝑔 ∶ ℝ → ℝbe a continuous function.
Let𝑓 ∶ ℝ → ℝbe a differentiable function.
Assume that∀𝑥 ∈ ℝ, 𝑓′(𝑥) = 𝑔(𝑥).
Which of the following statements must be true?
1 𝑓 (𝑥) =
∫
𝑥 0
𝑔(𝑡)𝑑𝑡.
2 If𝑓 (0) = 0, then𝑓 (𝑥) =
∫
𝑥 0
𝑔(𝑡)𝑑𝑡.
3 If𝑔(0) = 0, then𝑓 (𝑥) =
∫
𝑥 0
𝑔(𝑡)𝑑𝑡.
4 There exists𝐶 ∈ ℝsuch that𝑓 (𝑥) = 𝐶 +
∫
𝑥 0
𝑔(𝑡)𝑑𝑡.
5 There exists𝐶 ∈ ℝsuch that𝑓 (𝑥) = 𝐶 +
∫
𝑥 1
𝑔(𝑡)𝑑𝑡.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 3
True or False?
We want to find a function𝐻 with domainℝsuch that 𝐻(1) = −2and such that𝐻′(𝑥) = 𝑒sin𝑥for all𝑥.
Decide whether each of the following statements is true or false.
1 The function 𝐻(𝑥) =
∫
𝑥 0
𝑒sin𝑡𝑑𝑡 is a solution.
2 For any𝐶 ∈ ℝ, the function 𝐻(𝑥) =
∫
𝑥 0
𝑒sin𝑡𝑑𝑡 + 𝐶 is a solution.
3 There is a𝐶 ∈ ℝsuch that 𝐻(𝑥) =
∫
𝑥 0
𝑒sin𝑡𝑑𝑡 + 𝐶 is a solution.
4 The function 𝐻(𝑥) =
∫
𝑥 1
𝑒sin𝑡𝑑𝑡 − 2 is a solution.
5 There is more than one solution.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 4
A generalized version of FTC–1
Exercise
Let𝑓 be a continuous function with domainℝ.
Let𝑢,𝑣be differentiable functions with domainℝ.
Set
𝐻(𝑥) =
∫
𝑣(𝑥) 𝑢(𝑥)
𝑓 (𝑡)𝑑𝑡
Justify that𝐻 is differentiable onℝ and find a formula for 𝐻′(𝑥)
in terms of𝑓,𝑢,𝑣,𝑓′,𝑢′,𝑣′.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 5
An integral equation
Assume 𝑓 is a continuous function with domain ℝ that satisfies:
∀𝑥 ∈ ℝ,
∫
𝑥 0
𝑒
𝑡𝑓 (𝑡)𝑑𝑡 = sin 𝑥 𝑥
2+ 1 Find an explicit expression for 𝑓 (𝑥).
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 21, 2019 6
