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(1)

MAT137Y1 – LEC0501

Calculus!

The Fundamental Theorem of Calculus

Part 1

(2)

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)

(3)

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.

(4)

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.

(5)

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.

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An integral equation

Assume 𝑓 is a continuous function with domain ℝ that satisfies:

∀𝑥 ∈ ℝ,

𝑥 0

𝑒

𝑡

𝑓 (𝑡)𝑑𝑡 = sin 𝑥

𝑥

2

+ 1

Find an explicit expression for 𝑓 (𝑥).

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