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1.- Find f'(x) for the following functions. (b) f(x) = g(tan (x²)), where g is a differentiable function and n is a constant

1.- Find f'(x) for the following functions. (b) f(x) = g(tan (x²)), where g is a differentiable function and n is a constant

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Mathematical Problems for Calculus #### 1. Differentiation - Find \( f'(x) \) for the following functions. - (a) [Function not visible] - (b) \( f(x) = g^n(\tan(x^2)) \), where \( g \) is a differentiable function and \( n \) is a constant. #### 2. Integration Using Substitution - Find the following integrals using substitution. - (a) \( \int \tan^3(ax) \sec^2(ax) \, dx \), where \( a \) is a constant. - (b) \( \int \frac{3x^2}{1 + x^6} \, dx \), where \( a \) is a constant. #### 3. Volume of Solids - Let \( R \) be the region bounded by \( y = x^2 \), \( x = 1 \), and \( y = 0 \). Use the shell method to find the volume of the solid generated when \( R \) is revolved about the following lines. - (a) \( x = -2 \) - (b) \( x = 2 \) - (c) \( y = -2 \) - (d) \( y = 2 \) #### 4. Volume Calculation Methods - Pikachu claims that regardless of the type of problem you have (to find the volume), that you can always use either the washer or the shell method. Is she/he correct? Explain your reasoning.
Transcribed Image Text:### Mathematical Problems for Calculus #### 1. Differentiation - Find \( f'(x) \) for the following functions. - (a) [Function not visible] - (b) \( f(x) = g^n(\tan(x^2)) \), where \( g \) is a differentiable function and \( n \) is a constant. #### 2. Integration Using Substitution - Find the following integrals using substitution. - (a) \( \int \tan^3(ax) \sec^2(ax) \, dx \), where \( a \) is a constant. - (b) \( \int \frac{3x^2}{1 + x^6} \, dx \), where \( a \) is a constant. #### 3. Volume of Solids - Let \( R \) be the region bounded by \( y = x^2 \), \( x = 1 \), and \( y = 0 \). Use the shell method to find the volume of the solid generated when \( R \) is revolved about the following lines. - (a) \( x = -2 \) - (b) \( x = 2 \) - (c) \( y = -2 \) - (d) \( y = 2 \) #### 4. Volume Calculation Methods - Pikachu claims that regardless of the type of problem you have (to find the volume), that you can always use either the washer or the shell method. Is she/he correct? Explain your reasoning.
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