Chapter 4: Q. 1 (page 402)

Consider the differential equation $f' (x) = g (x)$ . What has to be true about $g (x)$ for the Second Fundamental Theorem of Calculus to guarantee that a function $f$ exists that satisfies $f' (x) = g (x)$ ?

Short Answer

Expert verified

For, $f' (x) = g (x)$ to be true two conditions need to be satisfied:

$F$ is continuous on $[a, b]$ and differentiable on $(a, b)$ .

$F$ is an antiderivative of $f$ , that is, $F' (x) = f (x)$ .

Step by step solution

Step 1. Given Information.

The objective to be true is to explain the second fundamental theorem of calculus.

Step 2. The conditions.

For, $f' (x) = g (x)$ to be true two conditions need to be satisfied:

$F$ is continuous on $[a, b]$ and differentiable on $(a, b)$ .

$F$ is an antiderivative of $f$ that is, $F' (x) = f (x)$ .

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Consider the differential equation $f' (x) = g (x)$ . What has to be true about $g (x)$ for the Second Fundamental Theorem of Calculus to guarantee that a function $f$ exists that satisfies $f' (x) = g (x)$ ?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The conditions.

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Consider the differential equation f'(x)=g(x). What has to be true about g(x)for the Second Fundamental Theorem of Calculus to guarantee that a function fexists that satisfies f'(x)=g(x)?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The conditions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Consider the differential equation $f' (x) = g (x)$ . What has to be true about $g (x)$ for the Second Fundamental Theorem of Calculus to guarantee that a function $f$ exists that satisfies $f' (x) = g (x)$ ?