At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Let's consider the pair of functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = e^x \)[/tex] over the interval [tex]\( 0 \leq x \leq 5 \)[/tex].
We want to determine the points at which the exponential function [tex]\( g(x) = e^x \)[/tex] grows faster than the quadratic function [tex]\( f(x) = x^2 \)[/tex]. In other words, we need to find the points where [tex]\( e^x \)[/tex] is greater than [tex]\( x^2 \)[/tex].
First, we need to evaluate both functions over the given interval. By examining the values of [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 5 \)[/tex]:
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 = 0 \quad \text{and} \quad g(0) = e^0 = 1 \][/tex]
Here, [tex]\( 1 > 0 \)[/tex].
- At [tex]\( x = 0.05 \)[/tex]:
[tex]\[ f(0.05) \approx 0.00255 \quad \text{and} \quad g(0.05) \approx 1.05 \][/tex]
Here, [tex]\( 1.05 > 0.00255 \)[/tex].
- At [tex]\( x = 0.10 \)[/tex]:
[tex]\[ f(0.10) \approx 0.0102 \quad \text{and} \quad g(0.10) \approx 1.11 \][/tex]
Here, [tex]\( 1.11 > 0.0102 \)[/tex].
Continuing this evaluation for several points within the interval, we observe:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1 \quad \text{and} \quad g(1) = e \approx 2.72 \][/tex]
Here, [tex]\( 2.72 > 1 \)[/tex].
- At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 4 \quad \text{and} \quad g(2) \approx 7.39 \][/tex]
Here, [tex]\( 7.39 > 4 \)[/tex].
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 9 \quad \text{and} \quad g(3) \approx 20.09 \][/tex]
Here, [tex]\( 20.09 > 9 \)[/tex].
- At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 16 \quad \text{and} \quad g(4) \approx 54.60 \][/tex]
Here, [tex]\( 54.60 > 16 \)[/tex].
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 25 \quad \text{and} \quad g(5) \approx 148.41 \][/tex]
Here, [tex]\( 148.41 > 25 \)[/tex].
By examining these values, we see a consistent pattern:
For all [tex]\( x \)[/tex] in the interval [tex]\( 0 \leq x \leq 5 \)[/tex], the values of [tex]\( g(x) = e^x \)[/tex] are greater than the values of [tex]\( f(x) = x^2 \)[/tex]. Hence, the exponential function [tex]\( g(x) = e^x \)[/tex] is consistently growing at a faster rate than the quadratic function [tex]\( f(x) = x^2 \)[/tex] over the entire interval [tex]\( 0 \leq x \leq 5 \)[/tex].
We want to determine the points at which the exponential function [tex]\( g(x) = e^x \)[/tex] grows faster than the quadratic function [tex]\( f(x) = x^2 \)[/tex]. In other words, we need to find the points where [tex]\( e^x \)[/tex] is greater than [tex]\( x^2 \)[/tex].
First, we need to evaluate both functions over the given interval. By examining the values of [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 5 \)[/tex]:
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 = 0 \quad \text{and} \quad g(0) = e^0 = 1 \][/tex]
Here, [tex]\( 1 > 0 \)[/tex].
- At [tex]\( x = 0.05 \)[/tex]:
[tex]\[ f(0.05) \approx 0.00255 \quad \text{and} \quad g(0.05) \approx 1.05 \][/tex]
Here, [tex]\( 1.05 > 0.00255 \)[/tex].
- At [tex]\( x = 0.10 \)[/tex]:
[tex]\[ f(0.10) \approx 0.0102 \quad \text{and} \quad g(0.10) \approx 1.11 \][/tex]
Here, [tex]\( 1.11 > 0.0102 \)[/tex].
Continuing this evaluation for several points within the interval, we observe:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1 \quad \text{and} \quad g(1) = e \approx 2.72 \][/tex]
Here, [tex]\( 2.72 > 1 \)[/tex].
- At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 4 \quad \text{and} \quad g(2) \approx 7.39 \][/tex]
Here, [tex]\( 7.39 > 4 \)[/tex].
- At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 9 \quad \text{and} \quad g(3) \approx 20.09 \][/tex]
Here, [tex]\( 20.09 > 9 \)[/tex].
- At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 16 \quad \text{and} \quad g(4) \approx 54.60 \][/tex]
Here, [tex]\( 54.60 > 16 \)[/tex].
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = 25 \quad \text{and} \quad g(5) \approx 148.41 \][/tex]
Here, [tex]\( 148.41 > 25 \)[/tex].
By examining these values, we see a consistent pattern:
For all [tex]\( x \)[/tex] in the interval [tex]\( 0 \leq x \leq 5 \)[/tex], the values of [tex]\( g(x) = e^x \)[/tex] are greater than the values of [tex]\( f(x) = x^2 \)[/tex]. Hence, the exponential function [tex]\( g(x) = e^x \)[/tex] is consistently growing at a faster rate than the quadratic function [tex]\( f(x) = x^2 \)[/tex] over the entire interval [tex]\( 0 \leq x \leq 5 \)[/tex].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
