Mat 331 Calculus III Final Exam 1: Suppose That F Is One-To-

Mat 331 Calculus Iiifinal Exam1 Suppose Thatfis One To Onef

Mat 331 Calculus Iiifinal Exam1 Suppose Thatfis One To Onef

MAT 331 CALCULUS III FINAL EXAM

1) Suppose that \(f\) is one–to–one, \(f(7) = 3\), and \(\lim_{x \to c} f(x) = L\). Find (a) \(f^{-1'}(3)\) and (b) \(\lim_{x \to c} f^{-1}(x)\).

2) Sketch the graph of the function \(y = \frac{2 + x}{-x}\).

3) Find the inverse function of \(f(x) = \frac{x + 1}{x - 1}\).

4) Solve the equation \(17 = x e^x\) for \(x\).

5) Differentiate:

  • (a) \(f(t) = t^2 \ln t\)
  • (b) \(g(t) = t e^t\)
  • (c) \(y = \arcsin x = \arctan y\)

6) Evaluate the limits:

  • (a) \(\lim_{x \to \infty} \frac{x}{e^x}\)
  • (b) \(\lim_{x \to 3} \frac{\arctan x}{x}\)

7) Evaluate \(\int_{a}^{b} t^2 + t \, dt\) by substituting \(t = 4u\) and changing the limits of integration.

8) Evaluate \(\int_{a}^{b} x^2 \, dx\). Use integration by parts to evaluate \(\int x^2 \tan x \, dx\). (Hint: let \(u = x\) and \(dv = \tan x \, dx\)).

9) Evaluate \(\int x^2 \, dx\). (Hint: let \(x = \sec \theta\) and change the limits of integration).

10) Evaluate \(\int_{a}^{b} \frac{q}{q^2 + 9} \, dq\).

Paper For Above instruction

The final exam for MAT 331 Calculus III encompasses a diverse set of fundamental topics in advanced calculus, requiring both theoretical understanding and practical problem-solving skills. This comprehensive assessment tests inverse functions, graphing, solving exponential equations, differentiation, limits, definite integrals, substitution, integration by parts, and trigonometric substitution techniques, providing a holistic measure of student proficiency in multivariable calculus concepts.

Beginning with the inverse functions, it is critical to recall that when \(f\) is one-to-one, its inverse function \(f^{-1}\) exists and is also continuous and differentiable within the domain where \(f\) is invertible. For problem 1, given \(f(7) = 3\) and the limit of \(f\) at some \(c\), we are to find the derivatives of the inverse at the corresponding points. Recognizing that \(\left(f^{-1}\right)'(y) = \frac{1}{f'(x)}\), where \(f(x) = y\), allows us to compute the derivatives provided the derivative of the original function is known at specific points. Additionally, the limit of the inverse function as \(x \to c\), is obtained by substituting the limit value accordingly, aligned with the inverse relationship.

Graphing the function \(y = \frac{2 + x}{-x}\) in problem 2 involves identifying asymptotes, intercepts, and the end behavior. Noticing that the function is a rational function, it has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y = -1\). The graph’s shape is typical of rational functions, with the domain excluding \(x=0\). Critical points can be located by setting numerator and denominator expressions to zero and inspecting the behavior near the asymptotes.

In problem 3, finding the inverse of \(f(x) = \frac{x + 1}{x - 1}\) requires solving for \(x\) in terms of \(y\) and then swapping variables. The inverse function \(f^{-1}(x)\) becomes \(\frac{x + 1}{x - 1}\) inverted to find \(x\) in terms of \(y\), leading to the inverse \(\boxed{f^{-1}(x) = \frac{x + 1}{x - 1}}\).

Problem 4 involves solving the transcendental equation \(17 = x e^x\). Since this typical form involves Lambert W function, recognizing that the solution can be expressed using \(x = W(17)\), where \(W\) is Lambert's W function, is crucial. Alternatively, numerical methods or iterative solutions can approximate the root with high precision.

For differentiation tasks in problem 5, applying the product rule for \(f(t) = t^2 \ln t\) in part (a), the chain rule for \(g(t) = t e^t\) in part (b), and the derivatives of inverse trig functions in part (c), ensure comprehensive understanding of differentiation rules. The derivatives are essential in analyzing the behavior of functions and in solving related problems involving rates and slopes.

Limit evaluations in problem 6 demonstrate how exponential growth dominates polynomial growth, leading to zero limits in part (a), whereas the limit of \(\arctan x / x\) as \(x \to 3\) in part (b) can be directly computed or approximated to assess asymptotic behavior.

In problems 7 and 8, the substitution method simplifies integrals with changing variables: \(t = 4u\) in problem 7 affects the bounds, while parts involving \(\tan x\) and \(\sec \theta\) leverage integration by parts and trigonometric substitution respectively, each crucial in handling integrals of complex functions.

Part 9 emphasizes substitution with the secant function, transforming the integral into a more manageable form, often simplifying the limits and integrand, which is a core technique in advanced calculus for evaluating challenging definite integrals.

Finally, the integration of rational functions in problem 10 involving \(q/(q^2 + 9)\) exemplifies the standard partial fraction approach, creating a path to straightforward integration using basic techniques.

Overall, mastering these problems involves a thorough understanding of calculus fundamentals, substitution methods, inverse functions, differentiation rules, and integral calculus techniques, which are essential skills in the advanced calculus curriculum and have extensive applications in scientific and engineering disciplines.

References

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