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### Chapter6_practice_S09

Course: MA 123, Fall 2008
School: Kentucky
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Word Count: 1293

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values, Extreme the Mean Value Theorem, curve sketching and concavity Chapter 6: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Extreme values problems on a closed interval [1]. Suppose f (t) = 4 t if t &lt; 4 t 4 if t 4. 2 Find the minimum of f (t) on the interval [0, 6]. (a) 0 (b) (c) 4 (d) 6 (e) 8 [2]. Let g(s) = (a) 1/3 s1 ....

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values, Extreme the Mean Value Theorem, curve sketching and concavity Chapter 6: Practice/review problems The collection of problems listed below contains questions taken from previous MA123 exams. Extreme values problems on a closed interval [1]. Suppose f (t) = 4 t if t < 4 t 4 if t 4. 2 Find the minimum of f (t) on the interval [0, 6]. (a) 0 (b) (c) 4 (d) 6 (e) 8 [2]. Let g(s) = (a) 1/3 s1 . Find the maximum of g(s) on the interval [0, 2]. s+1 (b) 0 (c) 1/3 (d) 2/3 (e) Neither the maximum nor the minimum exists on the given interval. t2 2t + 2 if t < 1 t3 if t 1. 0 [3]. Suppose f (t) = Find the minimum of f (t) on the interval [0, 2]. (a) 1 (b) (c) 1 (d) 2 (e) 8 [4]. Let f (x) = 3x2 + 6x + 4. Find the maximum value of f (x) on the interval [2, 1]. (a) 5 (b) 7 (c) 9 (d) 13 (e) 1 [5]. Let G(x) = (x 3) + 6 (x 3) + 6 (b) 1 if x 3 if x < 3. Find the minimum of G(x) on the interval [10, 10]. (a) 3 (c) 6 (d) 19 (e) 6 [6]. Let g(s) = (a) 1 1 . Find the maximum of g(s) on the interval [0, 2]. s+1 (b) 0 (c) 1 (d) 2 (e) Neither the maximum nor the minimum exists on the given interval. [7]. Find the minimum value of f (x) = x3 3x + 3 on the interval [2, 4]. (a) 2 (b) 1 (c) 0 (d) 1 88 (e) 2 [8]. Find the maximum of g(t) = |t + 4| + 10 on the interval [12, 12]. (a) 19 (b) 20 (c) 24 (d) 26 (e) 28 [9]. Find the minimum value of f (x) = (a) 1 (b) 2 x2 2x + 16 on the interval [0, 5]. (c) (d) 15 12 (e) 0 [10]. Let f (x) = |x2 1| + 2. Find the minimum of f (x) on the interval [3, 3]. (a) 3 (b) 0 (c) 1 (d) 2 (e) 1 [11]. Suppose f (t) = 2t3 9t2 + 12t + 31. Find the value of t in the interval [0, 3] where f (t) takes on its minimum. (a) 0 (b) 1 (c) 2 (d) 3 (e) Neither the maximum nor the minimum exists on the given interval. [12]. Let Q(t) = t2 . Find a value A such that the average rate of change of Q(t) from 1 to A equals the instantaneous rate of change of Q(t) at t = 2A (a) 1 (b) 1 3 (c) 1 4 (d) 1 5 (e) Does not exist Mean Value Theorem problems [13]. Find the value of A such that the average rate of change of the function g(s) = s3 on the interval [0, A] is equal to the instantaneous rate of change of the function at s = 1. (a) 2 (b) 3 (c) 5 (d) 6 (e) 12 [14]. Suppose k(s) = s2 + 3s + 1. Find a value c in the interval [1, 3] such that k (c) equals the average rate of change of k(s) on the interval [1, 3]. (a) 1 (b) 0 (c) 1 (d) 2 (e) 3 [15]. Let k(x) = x3 + 2x. Find a value of c between 1 and 3 such that the average rate of change of k(x) from x = 1 to x = 3 is equal to the instantaneous rate of change of k(x) at x = c. (a) 30 (b) 15 (c) 28 3 (d) 13 3 (e) 60 Increasing/decreasing problems [16]. Which function is always increasing on (0, 2) (a) x + x2 (d) 7 |x| 89 (b) x + (1/x) (e) (x 1)4 (c) x3 3x [17]. Suppose that a function f (x) has derivative f (x) = x2 + 1. Which of the following statements is true about the graph of y = f (x)? (a) The function is increasing on (, ) (b) The function is decreasing on (, ) The function is increasing on (, 0), and the function is decreasing on (0, ). (c) The function is increasing on (, 1) and (1, ), and the function is decreasing on (1, 1). (d) (e) The function is decreasing on (, 0), and the function is increasing on (0, ). [18]. Find the largest value of A such that the function g(s) = s3 3s2 24s + 1 is increasing on the interval (5, A). (a) 4 2 (b) 2 (c) 0 (d) 2 (e) 4 [19]. Let f (x) = ex . Find the intervals where f (x) is decreasing. (a) (, 0) (d) (1, ) (b) (0, ) (e) (1, 1) (c) (, 1) [20]. Let f (x) = x ln x. Find the intervals where f (x) is increasing. (a) (0, ) (d) (1/e, ) (b) (1, ) (e) (1/e, e) (c) (e, ) [21]. Suppose the cost, C(q), of stocking a quantity q of a product equals C(q) = 100 + q. The rate of change q of the cost with respect to q is called the marginal cost. When is the marginal cost positive? (a) q 10 > (b) q > 15 (c) q < 20 (d) q < 25 (e) q = 30 [22]. For which values of t is the function t3 2t + 1 increasing? (a) t > 2/3 or t < 2/3 (b) 2/3 < t < 2/3 (c) 0 < t < 4/3 (e) Never (d) 4/3 < t < 0 [23]. Suppose that g (x) = x2 x 6. Find the interval(s) where g(x) is increasing. (a) (1, 2) (d) (2, 3) (b) (, 2) and (3, ) (c) (, 1) and (2, ) (e) It cannot be determined from the information given [24]. Let f (x) = xe2x . Then f is decreasing on the following interval. (a) (, 1/2) (d) (1/2, ) 90 (b) (1/2, ) (e) (, 0) (c) (, 1/2) [25]. Find the interval(s) where f (x) = x3 + 18x2 105x + 4 is increasing. (Note that the coecient of x3 is 1, so compute carefully.) (a) (, 5) and (7, ) (d) (5, 7) (b) (5, 7) (e) (7, 5) (c) (, 5) and (7, ) [26]. Suppose that f (x) = xg(x), and for all positive values of x the function g(x) is negative (i.e., g(x) < 0) and decreasing. Which of the following is true for the function f (x)? (a) f (x) is negative and decreasing for all positive values of x. (b) (c) (d) f (x) is positive and increasing for all positive values of x. f (x) is negative and increasing for all positive values of x. f (x) is positive and decreasing for all positive values of x. (e) None of the above. [27]. Suppose the derivative of a function g(x) is given by g (x) = x2 1. Find all intervals on which g(x) is increasing. (a) (, ) (d) (0, ) (b) (1, 1) (e) (, 0) (c) (, 1) and (1, ) Extreme values problems using the rst derivative [28]. Suppose the derivative of the function h(x) is given by h (x) = 1 |x|. Find the value of x in the interval [1, 1] where h(x) takes on its minimum value. (a) 1/2 (b) 1 (c) 0 (d) 1/2 (e) 1 [29]. Suppose the total cost, C(q), of producing a quantity q of a product equals C(q) = 1000 + q + 1 2 q . 10 The average cost, A(q), equals the total cost divided by the quantity produced. What is the minimum average cost? (Assume q > 0) (a) 20 (b) 21 (c) 26 (d) 30 (e) 31 [30]. Suppose that a function h(x) has derivative h (x) = x2 + 4. Find the x value in the interval [1, 3] where h(x) takes its minimum. (a) 1 (b) 3 (c) 5 (d) 13 (e) 29 91 [31]. Suppose the cost, C(q), of stocking a quantity q of a product equals C(q) = value of q gives the minimum cost? (a) 10 (b) 15 (c) 20 ln x . x (b) (d) (d) 25 100 + q. Which positi...

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