Textbook Question
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = eˣ(x - 2)²
1
views
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.4.46
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = e⁻ˣ²/₂
Verified step by step guidance1
Identify the function: The given function is \( f(x) = e^{-x^2/2} \). This is a Gaussian function, which is a type of bell curve centered at the origin.
Determine the domain and range: The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \), because the exponential function is defined for all real numbers. The range is \( (0, 1] \) because \( e^{-x^2/2} \) is always positive and reaches a maximum value of 1 when \( x = 0 \).
Find the symmetry: The function \( f(x) = e^{-x^2/2} \) is an even function because \( f(-x) = f(x) \). This means the graph is symmetric with respect to the y-axis.
Determine the critical points: To find critical points, compute the derivative \( f'(x) \) and set it to zero. The derivative is \( f'(x) = -x e^{-x^2/2} \). Setting \( f'(x) = 0 \) gives \( x = 0 \) as the only critical point. Since \( f'(x) \) changes sign around \( x = 0 \), this is a maximum point.
Analyze the behavior at infinity: As \( x \to \pm\infty \), \( e^{-x^2/2} \to 0 \). This means the graph approaches the x-axis but never touches it, indicating horizontal asymptotes at \( y = 0 \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant. In the given function f(x) = e^(-x²/2), 'e' is the base of the natural logarithm, approximately equal to 2.71828. Understanding the behavior of exponential functions, especially with negative exponents, is crucial for graphing as they determine the function's growth or decay.
Recommended video:
6:13
Exponential Functions
Transformation of Functions
Transformations involve shifting, stretching, or compressing the graph of a function. For f(x) = e^(-x²/2), the exponent -x²/2 indicates a horizontal reflection and a vertical compression. Recognizing these transformations helps in predicting the shape and orientation of the graph, which is essential for accurate plotting.
Recommended video:
5:25Intro to Transformations
Critical Points and Symmetry
Critical points, where the derivative is zero or undefined, help identify local maxima, minima, or points of inflection. For f(x) = e^(-x²/2), symmetry about the y-axis is evident due to the even power of x. Analyzing these aspects allows for a more complete understanding of the function's behavior and assists in creating a detailed graph.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
g(s) = 1 / (s² + 1)
Textbook Question
Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).
Textbook Question
Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)
Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x/(x²+9)⁵ on [-2,2]
1
views
Textbook Question
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = x² - 10; x₀ = 3
1
views