Math 207 Final Review: Graph Of The Picture Showing Page 11

Math 207 Final Review Page 11 The Picture Shows The Graph Of A Functi

The assignment involves multiple calculus problems, including graph sketching, proofs, limits, derivatives, integrals, differential equations, optimization, and applications of calculus to real-world problems such as physics and economics. The core tasks are to analyze given functions, compute derivatives and integrals, apply theorems such as the Intermediate Value Theorem and Mean Value Theorem, and interpret the results in physical or applied contexts.

Specifically, the assignment requests:

• Sketching derivatives and integrals based on the graph of a function.
• Proving differentiation formulas such as the derivative of arcsine.
• State and explain fundamental theorems like the Intermediate Value Theorem.
• Calculate various limits involving algebraic, logarithmic, exponential, and trigonometric functions.
• Differentiate a series of functions, including logarithmic, exponential, trigonometric, and composite functions.
• Differentiate integral functions where the integrand depends on the variable of differentiation.
• Solve differential equations and find specific derivatives (second derivatives) given implicit relations.
• Verify the conditions of the Mean Value Theorem for particular functions and intervals.
• Calculate Riemann sums and exact definite integrals for given functions and partitions.
• Compute integrals involving rational functions, algebraic, exponential, logarithmic, and trigonometric functions.
• Determine the domain, invertibility, and derivatives of functions, along with inverse function formulas.
• Model physical phenomena such as the motion of a projectile, spherical growth, or water levels in a tank using calculus concepts.
• Solve optimization problems related to maximizing area, volume, or profit under constraints.
• Apply calculus to geometric problems such as tangent lines, maximum volume, or inscribed shapes.
• Model real-world scenarios like virus spread, economic costs and revenues, or circle growth rates.

Paper For Above instruction

This comprehensive analysis aims to address the multifaceted calculus problems outlined in the provided assignment. We begin by analyzing the graph of the function f, focusing on sketching the derivative g(x) = f'(x) and the integral F(x) where F’(x) = f(x). These foundational steps leverage the fundamental theorem of calculus and properties of derivatives and antiderivatives, linking the geometry of the graph to the calculus concepts involved.

Graph Sketching and Fundamental Calculus Concepts

Given the graph of a function f, sketching its derivative g(x) = f'(x) involves identifying critical points where the slope of f changes, maxima and minima, and points of inflection. These points translate into zeros, sign changes, and potential discontinuities in g(x). For example, a local maximum on f corresponds to g(x) crossing zero from positive to negative, while an inflection point correlates with a local extremum or zero of the second derivative.

Constructing F(x) where F’(x) = f(x) involves integrating the given function f. This process considers the area under the curve of f, modifying for initial conditions to determine the particular antiderivative. The graph of F reflects accumulated area, monotonically increasing where f is positive and decreasing where f is negative. Critical points of F occur where f(x) = 0.

Proving Differentiation Formula

The derivative of sin^–1 x, i.e., arcsine, is justified by implicit differentiation of the defining relation y = sin^–1 x, leading to dy/dx = 1/√(1 – x²). This proof employs differentiation rules and the Pythagorean identity sin² θ + cos² θ = 1.

The Intermediate Value Theorem

The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b], then for any value y between f(a) and f(b), there exists some c in (a, b) such that f(c) = y. This theorem ensures the existence of roots or certain function values within continuous intervals.

Limits Calculations

The assignment involves evaluating limits such as lim_x→5 (x² – 25)/(x² – 2x + 15), which simplifies through algebraic factorization to determine the behavior at the specified points. Limits involving logarithms, exponential functions, and trigonometric functions are computed using standard techniques including L'Hôpital's rule, conjugate multiplication, and Taylor expansions.

Differentiation of various functions

Differentiation of functions such as f(x) = ln(3√(x⁵) – 2x + 4) leverages chain rule and logarithmic differentiation. Derivatives of exponential functions like f(x) = 3e^4x2, and trigonometric functions like y = tan^–1 x⁴ are executed with appropriate rules, including product, quotient, and chain rule.

Differentiation involving integrals

When differentiating functions defined by integrals with variable limits, such as d/dx of ∫_x² T(t) dt, the Leibniz rule is applied. These calculations integrate the fundamental theorem of calculus with chain rule adaptations.

Solving Differential Equations and Second Derivatives

The second derivative ÿ can be determined from implicit relationships between x and y. For example, given x³ + y³ = 2, implicit differentiation yields ÿ, which informs about concavity and inflection points of the curve.

Application to Physical and Geometric Models

Modeling projectile motion on the Moon using initial velocity and gravitational acceleration involves integrating acceleration to find velocity, then integrating velocity to find position, and computing maximum height, time of flight, and impact velocity. Similarly, maximizing the area of a trapezoid with a fixed perimeter employs calculus-based optimization techniques.

Optimization and Geometry

Optimization problems, such as maximizing the volume of a cone inscribed in a sphere, involve expressing the volume as a function of a variable dimension and applying calculus to find critical points. The maximum volume is obtained at the critical point where the first derivative equals zero, and the second derivative test confirms whether it's a maximum.

Applications in Economics and Population Dynamics

Economic cost functions C(q) = q³ – 155q² + 6375q + 3000 are analyzed by differentiating to find fixed costs, minimal costs, and optimal production levels. In population models with logistic growth functions, the maximum infection rate occurs where the derivative g’(t) is maximized, typically at the inflection point of the spreading curve.

Conclusion

This extensive calculus review integrates core techniques of differentiation, integration, limits, and application models to provide a comprehensive understanding suitable for advanced mathematical analysis. Mastery of these topics enables solving complex real-world problems involving physics, economics, biology, and engineering, demonstrating the versatility of calculus concepts across disciplines.

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