https://file.aiquickdraw.com/mathbot/file/1730366693457y270u00x.jpg

Let's solve each question one at a time.

### Question 4
We are given:
\[
G(x) = \int_3^x (t + 1)^3 \, dt
\]
and asked to compute \( G'(7) \).

Using the **Fundamental Theorem of Calculus**:
\[
G'(x) = \frac{d}{dx} \int_3^x (t + 1)^3 \, dt = (x + 1)^3
\]
since differentiating the integral with respect to the upper limit \( x \) replaces \( t \) with \( x \) in the integrand.

Thus,
\[
G'(7) = (7 + 1)^3 = 8^3 = 512.
\]
**Answer**: \( G'(7) = 512 \).

---

### Question 5
We are asked to evaluate:
\[
\int_0^5 f(x) \, dx
\]
using the provided graph of \( f(x) \).

To do this, we will calculate the areas under the curve from \( x = 0 \) to \( x = 5 \), noting that areas above the x-axis are positive and areas below are negative.

1. **From \( x = 0 \) to \( x = 1 \)**: The area is a triangle with base 1 and height 1, so:
   \[
   \text{Area} = \frac{1}{2} \times 1 \times 1 = 0.5
   \]

2. **From \( x = 1 \) to \( x = 3 \)**: The area is a triangle with base 2 and height 2, so:
   \[
   \text{Area} = \frac{1}{2} \times 2 \times 2 = 2
   \]

3. **From \( x = 3 \) to \( x = 4 \)**: The area is a triangle below the x-axis with base 1 and height 1, so:
   \[
   \text{Area} = -\frac{1}{2} \times 1 \times 1 = -0.5
   \]

4. **From \( x = 4 \) to \( x = 5 \)**: The area is a triangle below the x-axis with base 1 and height 2, so:
   \[
   \text{Area} = -\frac{1}{2} \times 1 \times 2 = -1
   \]

Adding these areas together:
\[
\int_0^5 f(x) \, dx = 0.5 + 2 - 0.5 - 1 = 1
\]
**Answer**: \( \int_0^5 f(x) \, dx = 1 \).

---

Do you need further clarification on any step? 

### Related Questions
1. How would the solution to Question 4 change if the lower limit of the integral was not constant?
2. What is the significance of the Fundamental Theorem of Calculus in computing derivatives of integral functions?
3. How would the answer to Question 5 change if the graph had different values on the y-axis?
4. What method could we use to approximate integrals if we only had a table of values instead of a graph?
5. How does the concept of signed area apply to interpreting definite integrals with portions below the x-axis?

### Tip
For integrals involving graphs, always remember that areas below the x-axis contribute negatively to the integral.

Given the function G(x) = ∫ from 3 to x of (t + 1)^3 dt, compute G'(7). Given the function f(x) whose graph is provided, write the value of ∫ from 0 to 5 of f(x) dx.

Learn how to compute the derivative G'(7) for a given integral function G(x) using the Fundamental Theorem of Calculus. Additionally, evaluate the integral ∫ from 0 to 5 of f(x) dx based on the area under the graph of f(x). This tutorial covers essential calculus concepts and is ideal for students in undergraduate calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation