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1. For the (4x)dx, state the a) lower limit/bound b) upper limit/bound c) symbol for integral d) value of definite integral 7x+5 2. Find lim dx 3. For set up the partial fractions, but do not M(X² +4x+4) evaluate A, B, C.

1. For the (4x)dx, state the a) lower limit/bound b) upper limit/bound c) symbol for integral d) value of definite integral 7x+5 2. Find lim dx 3. For set up the partial fractions, but do not M(X² +4x+4) evaluate A, B, C.

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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### Calculus Practice Problems 1. For the integral \(\int_{0}^{2} (4-x) \, dx\), state the: - a) Lower limit/bound - b) Upper limit/bound - c) Symbol for integral - d) Value of definite integral 2. Find the limit: \[ \lim_{{x \to \infty}} \frac{7x + 5}{x^2 - 4x} \] 3. For the integral \(\int \frac{dx}{x(x^2 + 4x + 4)}\), set up the partial fractions, but do not evaluate A, B, C. ### Explanation of Concepts - **Definite Integral**: Evaluate the area under a curve from the lower limit to the upper limit. - **Limit**: Determine the behavior of a function as the input approaches a certain value. - **Partial Fractions**: Decompose a rational function into simpler fractions to integrate more easily. These problems are designed to reinforce understanding of integrals, limits, and partial fraction decomposition, foundational concepts in calculus.
Transcribed Image Text:### Calculus Practice Problems 1. For the integral \(\int_{0}^{2} (4-x) \, dx\), state the: - a) Lower limit/bound - b) Upper limit/bound - c) Symbol for integral - d) Value of definite integral 2. Find the limit: \[ \lim_{{x \to \infty}} \frac{7x + 5}{x^2 - 4x} \] 3. For the integral \(\int \frac{dx}{x(x^2 + 4x + 4)}\), set up the partial fractions, but do not evaluate A, B, C. ### Explanation of Concepts - **Definite Integral**: Evaluate the area under a curve from the lower limit to the upper limit. - **Limit**: Determine the behavior of a function as the input approaches a certain value. - **Partial Fractions**: Decompose a rational function into simpler fractions to integrate more easily. These problems are designed to reinforce understanding of integrals, limits, and partial fraction decomposition, foundational concepts in calculus.
### Calculus Exercises: Integration #### Problem 4: **Task:** Evaluate the integral \[ \int \frac{4x - 3}{x + 1} \, dx \] This is an integral where the numerator, \(4x - 3\), is a linear polynomial, and the denominator, \(x + 1\), is also linear. Consider using polynomial long division or substitution to simplify the integral for evaluation. #### Problem 5: **Task:** Evaluate the integral \[ \int \frac{x + 7}{(x - 6)(x - 2)} \, dx \] This integral involves a rational function where the denominator is the product of two linear factors: \( (x - 6) \) and \( (x - 2) \). Partial fraction decomposition is a useful technique here to split the function into simpler fractions that can be easily integrated. These integrals are typical exercises in a calculus course to practice integration techniques such as substitution, partial fraction decomposition, and polynomial long division.
Transcribed Image Text:### Calculus Exercises: Integration #### Problem 4: **Task:** Evaluate the integral \[ \int \frac{4x - 3}{x + 1} \, dx \] This is an integral where the numerator, \(4x - 3\), is a linear polynomial, and the denominator, \(x + 1\), is also linear. Consider using polynomial long division or substitution to simplify the integral for evaluation. #### Problem 5: **Task:** Evaluate the integral \[ \int \frac{x + 7}{(x - 6)(x - 2)} \, dx \] This integral involves a rational function where the denominator is the product of two linear factors: \( (x - 6) \) and \( (x - 2) \). Partial fraction decomposition is a useful technique here to split the function into simpler fractions that can be easily integrated. These integrals are typical exercises in a calculus course to practice integration techniques such as substitution, partial fraction decomposition, and polynomial long division.
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