CALCULUS+ITS APPLICATIONS
12th Edition
ISBN: 9780135164884
Author: BITTINGER
Publisher: PEARSON
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Chapter CR, Problem 68CR
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To Maximize: The function
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Chapter CR Solutions
CALCULUS+ITS APPLICATIONS
Ch. CR - Prob. 1CRCh. CR - Prob. 2CRCh. CR - Prob. 3CRCh. CR - Prob. 4CRCh. CR - Prob. 5CRCh. CR - Prob. 6CRCh. CR - Prob. 7CRCh. CR - Prob. 8CRCh. CR - Prob. 9CRCh. CR - Prob. 10CR
Ch. CR - Prob. 11CRCh. CR - Prob. 12CRCh. CR - Prob. 13CRCh. CR - Prob. 14CRCh. CR - Prob. 15CRCh. CR - Prob. 16CRCh. CR - Prob. 17CRCh. CR - Prob. 18CRCh. CR - Prob. 19CRCh. CR - Prob. 20CRCh. CR - Prob. 21CRCh. CR - Prob. 22CRCh. CR - Prob. 23CRCh. CR - Prob. 24CRCh. CR - Prob. 25CRCh. CR - Prob. 26CRCh. CR - Prob. 27CRCh. CR - Prob. 28CRCh. CR - Prob. 29CRCh. CR - Prob. 30CRCh. CR - Prob. 31CRCh. CR - Prob. 32CRCh. CR - Prob. 33CRCh. CR - Prob. 34CRCh. CR - Prob. 35CRCh. CR - Prob. 36CRCh. CR - Prob. 37CRCh. CR - Prob. 40CRCh. CR - Prob. 41CRCh. CR - Prob. 42CRCh. CR - Prob. 43CRCh. CR - Prob. 44CRCh. CR - Prob. 45CRCh. CR - Prob. 46CRCh. CR - Prob. 47CRCh. CR - Prob. 48CRCh. CR - Prob. 49CRCh. CR - Prob. 50CRCh. CR - Prob. 52CRCh. CR - Prob. 54CRCh. CR - Prob. 55CRCh. CR - Prob. 56CRCh. CR - Prob. 57CRCh. CR - Prob. 58CRCh. CR - Prob. 59CRCh. CR - Prob. 61CRCh. CR - Prob. 62CRCh. CR - Prob. 63CRCh. CR - Suppose the rate of change of y with respect to x...Ch. CR - Prob. 65CRCh. CR - Prob. 66CRCh. CR - Prob. 67CRCh. CR - Prob. 68CRCh. CR - Prob. 69CRCh. CR - Prob. 70CRCh. CR - Prob. 71CRCh. CR - Prob. 72CR
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- 3. Consider the sequences of functions f₁: [-π, π] → R, sin(n²x) An(2) n f pointwise as (i) Find a function ƒ : [-T,π] → R such that fn n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]? Justify your answer. [10 Marks]arrow_forward1. (i) Give the definition of a metric on a set X. [5 Marks] (ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4, d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer. = (iii) Consider a metric space (R, d.), where = [10 Marks] 0 if x = y, d* (x, y) 5 if xy. In the metric space (R, d*), describe: (a) open ball B2(0) of radius 2 centred at 0; (b) closed ball B5(0) of radius 5 centred at 0; (c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] [5 Marks] [5 Marks]arrow_forward(c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] 2. Let C([a, b]) be the metric space of continuous functions on the interval [a, b] with the metric doo (f,g) = max f(x)g(x)|. xЄ[a,b] = 1x. Find: Let f(x) = 1 - x² and g(x): (i) do(f, g) in C'([0, 1]); (ii) do(f,g) in C([−1, 1]). [20 Marks] [20 Marks]arrow_forward
- Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNEarrow_forward17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y ANarrow_forward(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Proarrow_forward
- Evaluate the integral using integration by parts. Sx² cos (9x) dxarrow_forwardLet f be defined as follows. y = f(x) = x² - 5x (a) Find the average rate of change of y with respect to x in the following intervals. from x = 4 to x = 5 from x = 4 to x = 4.5 from x = 4 to x = 4.1 (b) Find the (instantaneous) rate of change of y at x = 4. Need Help? Read It Master Itarrow_forwardVelocity of a Ball Thrown into the Air The position function of an object moving along a straight line is given by s = f(t). The average velocity of the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a. A ball is thrown straight up with an initial velocity of 128 ft/sec, so that its height (in feet) after t sec is given by s = f(t) = 128t - 16t². (a) What is the average velocity of the ball over the following time intervals? [3,4] [3, 3.5] [3, 3.1] ft/sec ft/sec ft/sec (b) What is the instantaneous velocity at time t = 3? ft/sec (c) What is the instantaneous velocity at time t = 7? ft/sec Is the ball rising or falling at this time? O rising falling (d) When will the ball hit the ground? t = sec Need Help? Read It Watch Itarrow_forward
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