Math 155 Final Exam Problem Points Score

Math 155 Final Examnameproblem Points Score1 132 133 124 125 126 127

Evaluate the following definite and indefinite integrals. If necessary, use substitution. Show all of your work.

(a) R 2t 1 2 5 t dt

(b) R 3 cos(x)e sin(x)+5 dx

(c) R p ⇡ 0 x sin(x 2 ) dx

(d) Use integration by parts to evaluate R 2xe 5x dx.

Find the solution of a discrete-time dynamical system for a yeast population y = 0.3 y with initial y0 = 2000. Determine the half-life when the population halves. Graph the updating function for a bacteria population with a piecewise rule, starting from b0 = 7, and analyze its long-term behavior. Find all equilibria for a population model m = m(3 - m) with stability analysis. Find critical points and extrema of the function f(t) = t^3 - t + 1 on [0, 3]. For a bacteria population Q(B) = B^2 e^{0.002B}, find critical points and identify local extrema. Analyze the function f(x) = 2x^2 + 3x^4 / (3 + x^6), including its leading behavior as x→0 and x→∞, and sketch its graph. Model drug absorption c(t) with a differential equation, estimate values using Euler’s method. Estimate total starch produced by a plant using Riemann sums, and find exact values. Model a tree’s height growth with a discrete system, and analyze fish growth with differential equations, calculating total length change and solving initial value problems.

Paper For Above instruction

The given set of problems encompasses a broad spectrum of calculus concepts, including integration techniques, differential equations, discrete-time dynamical systems, and function analysis. In this paper, I will systematically address each component, demonstrating the application of calculus principles to biological systems, physical models, and mathematical analysis to provide comprehensive solutions and insights.

Evaluation of Integrals and Substitution Methods

The first set of problems involves evaluating definite and indefinite integrals, often requiring substitution to simplify the integrand. For example, integrals involving products of functions like cos(x) and sin(x) can be approached using substitution ransforming one function into a different variable. Specifically, for the integral ∫ 3 cos(x) e^{sin(x)+5} dx, substituting u = sin(x) results in du = cos(x) dx, making the integral more manageable. The indefinite integral ∫ 2t dt straightforwardly yields t^2 + C, while the definite integral involving a quadratic expression of the form 2t with bounds can be integrated directly or via substitution if the integrand suggests it.

Another integral involving p = 0 and x sin(x^2) may involve substitution u = x^2, highlighting the importance of recognizing inner functions to simplify the integral. Such techniques are foundational in calculus, enabling the evaluation of complex integrals efficiently and accurately.

Application of Discrete-Time Dynamical Systems

The population models of yeast and bacteria involve discrete-time dynamical systems. For the yeast population y = 0.3 y, starting with y0 = 2000, the solution follows a geometric sequence, y = y0 * (0.3)^t, which demonstrates exponential decay. The half-life, corresponding to the time when the population reduces to 1000, can be found by solving (0.3)^t = 0.5, resulting in t = log(0.5) / log(0.3). This quantifies the rate at which the yeast population halves, an important measure in biological and chemical processes.

Similarly, the bacteria model involves a piecewise function b = (8/5) b if b ≤ 5; otherwise, b = b + 15. Graphing this cobweb model involves iteratively plotting b against b and drawing successive points to analyze stability and long-term behavior. Starting from b0 = 7, the cobweb diagram reveals oscillations or convergence, indicating the system's equilibrium points and stability criteria.

Stability and Equilibria in Population Models

The model for moose population m = m(3 - m) with parameter h > 0 involves finding equilibrium points by setting m = m. Equilibria occur at m = 0 and m = 3, with potential for multiple solutions depending on h. Using the Stability Theorem, the derivative of the function at equilibrium determines stability: a magnitude less than 1 indicates stability. Specifically, at m = 0, the derivative is f'(0) = 3, which is > 1, indicating instability. At m = 3, f'(3) = -3, which in absolute value exceeds 1, suggesting instability as well; however, the stability depends on h's influence on the derivatives.

Function Analysis and Critical Points

The analysis of f(t) = t^3 - t + 1 involves finding critical points by setting f'(t) = 3t^2 - 1 = 0, which yields t = ± 1/√3. The second derivative, f''(t) = 6t, helps determine the nature of these points. On [0, 3], the maximum and minimum are identified by evaluating f(t) at critical and boundary points, confirming the global extremum placements.

The function Q(B) = B^2 e^{0.002B} has a critical point at B where its derivative equals zero. Calculating Q'(B), setting to zero, and solving yields a positive critical point B ≈ 100, which the second derivative test confirms as a maximum, indicating an optimal bacterial population for maximum Q.

Behavior of Rational Functions and Graphing

The function f(x) = (2x^2 + 3x^4) / (3 + x^6) exhibits specific asymptotic behaviors. As x → 0, f(x) ≈ (2x^2) / 3, dominated by the quadratic term, representing the leading behavior f0(x). As x → ∞, the dominant term in numerator and denominator is x^6, yielding f(x) ≈ 3x^4 / x^6 = 3 / x^2, tending to zero. Using matched leading behaviors helps sketch the graph, combining asymptotic analysis with the function's shape for a realistic depiction.

Differential Equations for Drug Absorption and Biological Modeling

The differential equation dc/dt = rate of drug entering the bacterium, with c(0) = 0.2 mol, models drug concentration over time. Applying Euler’s method with step size 0.5 estimates c(t) at larger t, making iterative calculations based on the current value and the derivative approximation.

Similarly, the plant’s starch production rate dS/dt = 4t / (1 + t^2) involves Riemann sums to estimate the total change over a specific interval, and the exact integral evaluates via substitution. The average rate of production over the period is obtained by dividing the total change by the interval length.

The growth of a tree with height h = 3 + 2t can be modeled as a discrete system h = h + 2, with initial height 3m. The fish growth differential equation dL/dt = 5.0 e^{0.2t} describes exponential growth, leading to an integral computation for total length increase and a solution of the initial value problem to determine L(t).

Conclusion

Overall, these problems demonstrate the broad application of calculus in modeling biological systems, analyzing functions, and solving real-world problems involving integration, differential equations, and system stability. Mastery of these concepts allows for effective analysis and prediction of complex phenomena across scientific disciplines.

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