Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = e⁻ˣ²/₂
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 4.3.82
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
Verified step by step guidance1
First, find the first derivative of the function f(x) = e^x(x - 2)^2. Use the product rule: if u(x) = e^x and v(x) = (x - 2)^2, then f'(x) = u'(x)v(x) + u(x)v'(x).
Calculate the derivatives: u'(x) = e^x and v'(x) = 2(x - 2). Substitute these into the product rule to find f'(x).
Set the first derivative f'(x) equal to zero to find the critical points. Solve the equation for x to find the values where the slope of the tangent is zero.
Next, find the second derivative f''(x) by differentiating f'(x) again. This will involve using the product rule and chain rule as necessary.
Evaluate the second derivative at each critical point found. Use the Second Derivative Test: if f''(x) > 0, the point is a local minimum; if f''(x) < 0, the point is a local maximum. If f''(x) = 0, the test is inconclusive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points of a function occur where its first derivative is zero or undefined. These points are essential for identifying potential local maxima and minima, as they represent locations where the function's slope changes. To find critical points, one must differentiate the function and solve for the values of x that satisfy the condition f'(x) = 0.
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Critical Points
Second Derivative Test
The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. It involves evaluating the second derivative of the function at the critical points. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum; and if f''(x) = 0, the test is inconclusive.
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06:02The Second Derivative Test: Finding Local Extrema
Exponential and Polynomial Functions
The function f(x) = eˣ(x - 2)² combines an exponential function, eˣ, and a polynomial function, (x - 2)². Understanding the behavior of these types of functions is crucial, as exponential functions grow rapidly, while polynomial functions can have varying degrees of growth based on their degree. This knowledge helps in analyzing the overall shape and critical points of the function.
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6:13Exponential Functions
Related Practice
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]
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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
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Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x³ + 3x² / 2 - 2x
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) = x ln (x + 1) -1 ; x₀ = 1.7
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) = (4x³/3) + 5x² - 6x on [0,5]
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