Textbook Question
Suppose f is differentiable on (-∞,∞) and the equation of the line tangent to the graph of f at x = 2 is y = 5x -3. Use the linear approximation to f at x = 2 to approximate f(2.01).
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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 4.9.17
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
ƒ(y) = - 2/y³
Verified step by step guidance1
Step 1: Recall the general rule for finding antiderivatives. The antiderivative of a function is the reverse process of differentiation. For power functions, the antiderivative of yⁿ is (y^(n+1))/(n+1), provided n ≠ -1.
Step 2: Rewrite the given function ƒ(y) = -2/y³ using exponents. Since 1/y³ is equivalent to y⁻³, the function becomes ƒ(y) = -2y⁻³.
Step 3: Apply the antiderivative rule to -2y⁻³. Increase the exponent by 1 (from -3 to -2) and divide by the new exponent (-2). This gives the antiderivative as (-2 * y⁻²)/(-2).
Step 4: Simplify the expression obtained in Step 3. The -2 in the numerator and denominator cancel out, leaving y⁻² or 1/y². Add the constant of integration, C, to account for all antiderivatives. The result is F(y) = 1/y² + C.
Step 5: Verify your work by differentiating F(y). Differentiate F(y) = 1/y² + C using the power rule and confirm that the derivative is ƒ(y) = -2/y³, which matches the original function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative
An antiderivative of a function is another function whose derivative is the original function. In calculus, finding antiderivatives is essential for solving problems related to integration. The process involves determining a function F(y) such that F'(y) = f(y). Antiderivatives are not unique; they can differ by a constant, represented as F(y) + C, where C is any constant.
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Power Rule for Integration
The Power Rule for Integration is a fundamental technique used to find antiderivatives of polynomial functions. It states that the integral of y^n, where n is not equal to -1, is (y^(n+1))/(n+1) + C. This rule simplifies the process of finding antiderivatives by allowing us to easily handle terms with exponents, making it particularly useful for functions like f(y) = -2/y³.
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Verification by Differentiation
Verification by differentiation involves checking the correctness of an antiderivative by taking its derivative. If the derivative of the antiderivative matches the original function, the solution is confirmed. This step is crucial in calculus to ensure that the integration process was performed correctly and helps reinforce the relationship between differentiation and integration.
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Related Practice
Textbook Question
Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.
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Textbook Question
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = x⁵/5 - x³/4 - 1/20
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Maximum-area rectangles Of all rectangles with a fixed perimeter of P, which one has the maximum area? (Give the dimensions in terms of P.)
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