Find The Intervals Where Is Increasing And The Int

Find The Intervals Whereis Increasing And The Int

1. Find the interval(s) where a given function is increasing and the interval(s) where it is decreasing.

2. Let \( f(x) = 2x^3 + x^2 - x \). (a) Find the interval(s) where \( f(x) \) is concave upward. (b) Find the interval(s) where \( f(x) \) is concave downward. (c) Find the x-coordinate(s) of any point(s) of inflection.

3. Sketch the graph of the function \( y = x^2 - 3x + 2 \).

4. A farmer wants to create a pen inside a barn, using two sides of the barn as part of the pen, and fencing material for the other two sides. The total fencing available is 80 feet. Find the maximum area that can be enclosed by the pen.

5. To make a rectangular trough, sides are bent from a 32-inch wide sheet of metal to form a specific cross section. Determine the dimensions of the cross section that maximize the area (and thus the volume of the trough).

6. Simplify the following expressions: (a) \( \frac{1}{x} - \frac{1}{x^2} \). (b) \( a^a - a^{-a} \).

7. Use the laws of logarithms to expand and simplify \( \log( x^3 + x) \).

8. A quantity \( Q(t) \) grows exponentially over time \( t \) according to \( Q(t) = Q_0 e^{kt} \), where \( t \) is in minutes. (a) What is the initial quantity \( Q_0 \)? (b) What is the quantity after 50 minutes?

9. Find the derivative of \( f(x) = e^{x+1} \).

10. Use logarithmic differentiation to find the derivative of \( y = x^x \). Also, find the indefinite integral of \( x^x \) with respect to \( x \).

11. Find the indefinite integral \( \int \frac{1}{x^2 - x} dx \).

12. Evaluate the definite integral \( \int_{a}^{b} \frac{x}{x^2 + 1} dx \).

13. The demand function for a product is \( p(x) = \frac{10}{x^2 - 0.01} \), where \( p \) is the unit price in dollars and \( x \) is the quantity demanded in thousands. Determine the consumer surplus when the price is set at $2.

14. Calculate the amount accumulated in an account if $800 is deposited monthly for 20 years at a 7% annual interest rate compounded continuously.

15. Evaluate the indefinite integral \( \int (6t^2 - 2t + 1) dt \).

16. Use a table of integrals to evaluate \( \int (2 \ln 4x) dx \).

17. Approximate \( \int_{0}^{b} \frac{dx}{x^2 + 1} \) using the Trapezoidal Rule with \( n = 6 \) subdivisions.

18. Approximate \( \int_{0}^{2} e^{x} dx \) using Simpson’s Rule with \( n = 8 \) subdivisions.

19. A scholarship fund will award $1000 annually, with interest compounded continuously at 7.5%. Determine the required initial endowment to sustain this fund.

Paper For Above instruction

The analysis of functions and their properties is a fundamental aspect of calculus, providing essential insights into the behavior of mathematical models across various disciplines. Understanding where a function is increasing or decreasing allows analysts to interpret trends, optimize processes, or identify local maxima and minima critical in fields like economics, engineering, and physical sciences. Additionally, the study of concavity and inflection points reveals the nature of a function’s curvature, aiding in understanding the acceleration of changes—crucial for modeling dynamic systems.

Intervals of Increase and Decrease

To find where a function is increasing or decreasing, the first derivative \( f'(x) \) must be examined. The critical points, where \( f'(x) = 0 \) or is undefined, partition the domain into intervals. Within each interval, the sign of \( f'(x) \) indicates whether the function is increasing (positive derivative) or decreasing (negative derivative). For example, considering the function \( f(x) = 2x^3 + x^2 - x \), computing the derivative yields \( f'(x) = 6x^2 + 2x - 1 \). Solving \( 6x^2 + 2x - 1 = 0 \), the critical points are \( x = \frac{-1 \pm \sqrt{1 + 6}}{6} \), which simplifies to \( x = \frac{-1 \pm \sqrt{7}}{6} \). The sign analysis around these critical points reveals the intervals where \( f(x) \) is increasing or decreasing.

Concavity and Points of Inflection

The second derivative \( f''(x) \) determines the concavity of \( f(x) \). When \( f''(x) > 0 \), the graph is concave upward; when \( f''(x) -\frac{1}{6} \), \( f''(x) > 0 \), indicating upward concavity. The inflection point occurs where the concavity changes sign, at \( x = -\frac{1}{6} \).

Graph Sketching and Application Problems

The quadratic function \( y = x^2 - 3x + 2 \) can be sketched by identifying its vertex at \( x = \frac{3}{2} \), calculating \( y \) at key points, and noting its opening upward. Optimization problems, such as fencing or trough dimensions, involve expressing the area or volume as functions of variables, differentiating to find maxima or minima, and verifying these points correspond to valid solutions within constraints.

For example, maximizing fencing within an 80-foot constraint involves setting the total fencing as an expression in terms of one variable—say, \( x \)—then differentiating, setting the derivative to zero, and solving for \( x \). This yields the dimensions that maximize enclosure area. Similarly, designing the largest-volume trough from a metal sheet involves expressing volume as a function of cross-section dimensions, differentiating, and solving for critical points.

Simplification and Logarithmic Techniques

Algebraic manipulations and the laws of logarithms facilitate simplifying complex expressions. For example, to simplify \( (x^a)^b \), one applies the rule \( (x^a)^b = x^{ab} \). Applying logarithmic properties, like \( \log(ab) = \log a + \log b \), can expand and simplify expressions like \( \log(x^3 + x) \), which assists in integration and differentiation processes.

Growth Functions and Derivatives

Exponential growth functions, such as \( Q(t) = Q_0 e^{kt} \), model processes where quantities increase at a rate proportional to their current value. The initial quantity \( Q_0 \) is directly obtained by evaluating \( Q(0) \). After 50 minutes, the quantity is \( Q(50) = Q_0 e^{50k} \). Derivatives of exponential functions, like \( f(x) = e^{x+1} \), are straightforward, as \( f'(x) = e^{x+1} \).

Logarithmic differentiation becomes essential for functions where variables are both in the base and exponent, such as \( y = x^x \). Differentiating such functions involves taking the natural log of both sides and applying implicit differentiation, which simplifies the differentiation process for complex variable-exponent functions.

In integration, indefinite integrals like \( \int \frac{1}{x^2 - x} dx \) involve partial fraction decomposition to rewrite the integrand in manageable parts, facilitating straightforward integration. Definite integrals evaluate the net accumulation of the function over an interval, crucial in calculating total quantities such as consumer surplus or accumulated interest.

Using numerical methods like the Trapezoidal and Simpson’s rules allows approximate estimation of integrals where analytical solutions are complicated or impossible. These methods employ subdividing the interval into smaller segments and summing weighted function evaluations to approximate the integral value accurately.

The applications of these calculus concepts extend to financial calculations such as endowment requirements for scholarships, where the present value must grow at a continuous rate to sustain annual disbursements. Formulas derived from exponential growth models help determine initial investments needed to fulfill future obligations efficiently.

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