https://file.aiquickdraw.com/mathbot/file/1726739015667cxxm3cgm.jpg

In this task, you are given two functions, \( g(x) = 4 \cdot (1 - e^{-x}) \) and \( h(x) = e^x - 1 \), and asked to calculate the area between these two curves and to demonstrate that the integral (antiderivative) of \( g(x) \) is increasing on the interval \( [0, \infty[ \).

### Part a) Finding the Area of \( M \)

The area \( M \) is the region bounded by the graphs of \( g(x) \) and \( h(x) \) in the first quadrant. The first step is to find the intersection point(s) of the two functions by solving \( g(x) = h(x) \):

\[
4 \cdot (1 - e^{-x}) = e^x - 1
\]

After determining the points of intersection, the area between the curves is found by integrating the difference between the two functions within the limits of integration (i.e., the intersection points).

\[
\text{Area} = \int_{x_1}^{x_2} (g(x) - h(x)) \, dx
\]

### Part b) Proving that the antiderivative of \( g(x) \) is increasing

To show that the antiderivative (integral) of \( g(x) \) is increasing on \( [0, \infty[ \), you need to prove that \( g(x) \geq 0 \) for all \( x \in [0, \infty[ \). Since \( g(x) \) represents the derivative of the antiderivative, the positivity of \( g(x) \) on this interval implies that its integral is increasing.

I'll now calculate the details for both parts. Let me start by solving the intersection point for part a).The functions \( g(x) = 4 \cdot (1 - e^{-x}) \) and \( h(x) = e^x - 1 \) intersect at \( x = 0 \) and \( x = \ln(4) \).

Now, to calculate the area between the curves from \( x = 0 \) to \( x = \ln(4) \), we will integrate the difference \( g(x) - h(x) \) over this interval:

\[
\text{Area} = \int_0^{\ln(4)} (g(x) - h(x)) \, dx
\]

I'll compute this area next.The area of the region \( M \) bounded by the curves \( g(x) \) and \( h(x) \) in the first quadrant is:

\[
\text{Area} = 5 \ln(4) - 6
\]

### Part b) Showing that the Antiderivative of \( g(x) \) is Increasing

To prove that the antiderivative of \( g(x) \) is increasing on the interval \( [0, \infty[ \), we need to show that \( g(x) \geq 0 \) for all \( x \geq 0 \). I will now check if \( g(x) \geq 0 \) on this interval.At \( x = 0 \), \( g(0) = 0 \), and as \( x \to \infty \), \( g(x) \to 4 \). Since \( g(x) \geq 0 \) for all \( x \geq 0 \), the function is non-negative on this interval. Therefore, the antiderivative of \( g(x) \) is indeed increasing on \( [0, \infty[ \).

Would you like more details or have any questions? Here are five related questions to expand the discussion:

1. How would the result change if the functions intersected at different points?
2. What is the significance of the area between curves in applications?
3. How can we check the monotonicity of other functions' antiderivatives?
4. Can the method of integration be applied to more complicated functions?
5. How does changing the limits of integration affect the area calculation?

**Tip**: When solving for intersections of functions, always check for extraneous solutions that might not lie within the desired interval.

To funktioner g og h er givet ved g(x) = 4 * (1 - e^-x) og h(x) = e^x - 1. Grafen for g og h afgrænser i første kvadrant en punktmængde M, der har et areal. a) Bestem arealet af M. b) Gør rede for, at stamfunktionen til g er voksende i [0; ∞[.

Learn how to calculate the area between the functions g(x) = 4(1 - e^-x) and h(x) = e^x - 1 in the first quadrant. Also, prove that the antiderivative of g(x) is increasing on the interval [0, ∞). This calculus problem is ideal for students in Grade 12 or 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