Calculus I Practice Exam 3B

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1 Calculus I Practice Exam 3B This practice exam emphasizes conceptual connections and understanding to a greater degree than the exams that are usually administered in introductory single-variable calculus courses. It is designed to guide students who are taking such courses to a deeper mastery of the material. While a number of questions here are fairly typical for actual examinations, you should not infer from the expression practice exam that exams encountered in introductory singlevariable calculus course swill ask the same types of questions. Multiple choice 5 1. Evaluate the integral below by interpreting it in terms of areas. 5 x 5 dx. A) 5π B) 5π C) 5π D). Find the absolute maximum of the function f(x) = 1 1 on the interval [, 1]. Think about x 1 the definition of absolute maximum carefully before you pick an answer. A) does not exist B) 3 C) 1 D) For the function f(x) = ln x on the interval [1, e], find the value of c that satisfies the conclusion of the Mean Value Theorem. 1 A) e 1 B) C) e 1 e 1 D) 1 ln 1 E) 1+e ln (e 1) e 1. Find the inflection values (x values of the inflection points) of f(x) = x e 3x. A), ±.77 B), ± C), ± 1 D), ± Consider the quantity S = e 1 + e + e 3 + e and the following statements: i. S is a left sum for the function f(x) = e x on the interval [,]. ii. S is a left sum for the function f(x) = e x on the interval [1,5]. iii. S is a right sum for the function f(x) = e x on the interval [,]. iv. S is a right sum for the function f(x) = e x on the interval [1,5]. A) All four statements are correct. B) The first and fourth statement are correct, the others incorrect. C) The second and third statement are correct, the others incorrect. D) They are all wrong because S is a left or right sum for e x on the interval [1,]. E) They are all wrong, but for a different reason than the one given in D). 6. If f(x)dx = and (f(x) + g(x))dx =, what is the value of g(x)dx? A) - B) C) -1 D) 1 16 R. Boerner, ASU School of Mathematical and Statistical Sciences 1

2 7. Find the closest distance d a point on the parabola y = x can have to the point (,1). A) d = 1 B) d = 3 C) d = 3 D) d = 1 E) None of these choices 8. When you lean a ladder against a wall, what base angle will maximize the area of the triangle formed by the ladder, the ground and the wall? A) B) π 6 C) π D) π 3 9. A twice differentiable function defined on the interval (, ) has a single critical point at x = 1. Furthermore, f (1) =. Which of the following conclusions can you draw? If there is more than one valid conclusion, select the strongest, i.e. the one that implies the other(s). A) f has a relative minimum at x=1. B) f has a relative maximum at x=1. C) f has an absolute minimum at x=1. D) f has an absolute maximum at x=1. E) f may not have any extrema of any kind. 1. Find an antiderivative F(x) of f(x) = x+1 defined on (, ) that satisfies F(1) =. A) F(x) = x +x 1 x B) F(x) = x +x + 1 x C) F(x) = x + 1 ln x D) F(x) = x + 1 ln x 1 Free response 1. Suppose that the second derivative of f is f (x) = (x 1)(x 3). x Carry out the concavity test to determine the intervals on which f is concave up and the intervals on which f is concave down and identify the x values of the inflection point(s). Intervals: x value for testing: f (x) is on this interval f is on this interval Inflection point(s): x=. Consider the function f(x) = sin x cos x + sin x on [,π]. Find the absolute maximum and absolute minimum and the location of each and justify your answers by showing appropriate work. Give exact answers, not calculator approximations. 16 R. Boerner, ASU School of Mathematical and Statistical Sciences

3 3. Below, you find the graph of the derivative of f. Based on this graph, find the intervals where f is increasing and decreasing, and the relative extrema. Complete the explanations by putting a single word into each blank. f is increasing on the interval(s) because f is there. f is decreasing on the intervals(s) because f is there. f has (a) relative maximum/maxima at x= because f changes from to there. f has (a) relative minimum/minima at x= because f changes from to there. 16 R. Boerner, ASU School of Mathematical and Statistical Sciences 3

4 Answers Calculus I Practice Exam 3B Multiple Choice: 1B C 3A C 5C 6C 7C 8C 9C 1D Free Response: 1. Since f (x) = (x 1)(x 3) is already given, we do not need to compute it and proceed directly to the concavity test. Since f (x) is continuous, it can only change sign where it is zero, and f (x) = for x = 1, 3. Intervals: (, 1) (1,3) (3, ) x value for testing: f (x) is on this interval negative positive positive f is on this interval concave down concave up concave up Inflection point(s): x=1 (Observe that there is no inflection point at x=3 because f does not change sign.). First we have to find the critical points of f(x) = sin x cos x + sin x. The derivative is f (x) = cos x sin x + cos x. To simplify this, we use the Pythagorean identity sin x = 1 cos x: f (x) = cos x + cos x 1. Since f is defined for all x in the domain [,π], f can only have critical points where f (x) =. We solve that equation by factoring: f (x) = cos x + cos x 1 = ( cos x 1)(cos x + 1) Therefore, the critical points of f are where cos x = 1 or where cos x = 1. Using the unit circle, we can find the solutions of those equations in the interval [,π]. The solutions of the first equation are x = π and x = 5π, and the solution of the second 3 3 equation is x = π. 16 R. Boerner, ASU School of Mathematical and Statistical Sciences

5 We now make a table of values of f at the critical values and the interval end points, using the known values of sin and cos on the unit circle. x f(x) π π 5π π We conclude that the absolute maximum of f on the interval [,π] is 3 3, and occurs at x = π 3 3, and that the absolute minimum of f on the interval [,π] is, at x = 5π f is increasing on the intervals (1,), (3,) because f is positive there. f is decreasing on the intervals (,1), (,3) because f is negative there. f has a relative maximum at x= because f changes from positive to negative there. f has relative minima at x= 1, 3 because f changes from negative to positive there. Note: while this guide is being made freely available to ASU students and the general public for personal use, it is not to be uploaded to third-party websites, especially not ones that profit from such content. If you found this document on a third-party website such as Course Hero or Chegg, the document is being served to you in violation of copyright law. 16 R. Boerner, ASU School of Mathematical and Statistical Sciences 5

Calculus I Practice Final Exam B

Calculus I Practice Final Exam B This practice exam emphasizes conceptual connections and understanding to a greater degree than the exams that are usually administered in introductory single-variable