Calculus II - Problem Solving Drill 23: Graphing Utilities in Calculus

Transcription

1 Calculus II - Problem Solving Drill 3: Graphing Utilities in Calculus Question No. 1 of 10 Question 1. Find the approximate maximum point of y = x + 5x + 9. Give your answer correct to two decimals. Question #01 (A) ( , 1.15) (B) (1.5,1.13) (C) (1.50,1.15) (D) (1.3,1.1) (E) (1.500,1.15) The precision of this answer is not correct. Indeed, you are asked to provide an answer correct to decimal places. B. Correct! The given function is a parabola which is concave down. You may see this by graphing the function. Thus it admit a maximum at the point x such that y (x) =0. The precision of this answer is not correct. Indeed, you are asked to provide an answer correct to decimal places. The precision of this answer is not correct. Indeed, you are asked to provide an answer correct to decimal places. E. Incorrect! The precision of this answer is not correct. Indeed, you are asked to provide an answer correct to decimal places. The given function is a concave down parabola. Therefore, it admits a maximum which is attained at the point x such that y (x) = 0. But y = -4x+5. Thus the maximum is attained at x=5/4. The valuation of the function at x=5/4 gives After rounding the answer to two decimals So the coordinates of the maximum point correct to two decimals are (1.5,1,13). By using a graphing aid having appropriate commands for finding the maximum, the final screenshot may look like this: Therefore the maximum point correct to two decimals is (1.5, 1.13).

2 Question No. of 10 Question. Find an approximation of 1 x 0 e dx correct to 4 decimals. Question #0 (A) (B) (C) (D) (E) We expect a positive answer since the integrand is positive and the limits of integration are increasing. We expect a positive answer since the integrand is positive and the limits of integration are increasing. C. Correct! By elimination, this answer is the only plausible one. Using numerical schemes such as the Midpoint Rule, the Trapezoidal Rule or Simpson Rule with a large enough number of subdivision, we arrive to the answer. This answer is not rounded to 4 decimals as requested. E. Incorrect This answer is not rounded to 4 decimals as requested. Several graphing aids have numerical integration procedure already programmed. Find the appropriate keystrokes for integration on your calculator and truncate the answer appropriately. The viewing window may look like this Therefore an approximation of the define integral correct to 4 decimals is

3 Question No. 3 of 10 Question 3. Find the approximate minimum value correct to 3 decimals for the function 10x 3 = x + 1 y. Question #03 (A) (B) (C) (D) -6.7 (E) The precision is not correct. The precision is not correct The precision is not correct. The precision is not correct E. Correct! Employ the first or second derivative test. Alternatively, you may graph the function using your graphing aid and use appropriate keystrokes to find the minimum. Graph the function using your graphing utility and employ the appropriate keystrokes to obtain the minimum. Alternatively, you may use the first or second derivative test. In case you employ a calculator the final screenshot may look like this: Therefore the minimum value correct to 3 decimals is

4 Question No. 4 of 10 Question 4. Approximate 1 x 3 1/ (1 + ) dx and provide an answer correct to 4 decimals 0 Question #04 (A) (B) (C) (D) (E) For x between 0 and 1, the integrand is bigger than. By integration from 0 to 1, we infer that the integral is bigger than. B. Correct! Use numerical scheme such as the Simpson Rule or employ numerical integrations schemes of your graphing aid.. The format of this answer is not correct. The precision of this answer is not correct. E. Incorrect The integral given is an area. So it must be positive Solve this problem by using the numerical integration procedure built in you calculator. The last screen may look like this: Thus the given integral is almost

5 Question No. 5 of 10 Question #05 Question 5. Approximate (A) (B) 0.93 (C) 0.99 (D) 0.99 (E) 0.9 cos(x ) dx and provide an answer correct to two decimals. The precision is wrong. The precision is not correct.. The precision is not correct The precision is not correct. E. Correct! The given integrand does not admit an anti-derivative in terms of elementary functions. So you must employ a numerical integration procedure from your graphing aid to arrive at the answer. Employ numerical integration procedures from your graphing aid. Here you may not use the Fundamental Theorem of Calculus since the integrand does not have an ant-derivative expressible in terms of elementary functions. Using graphing utilities integration procedure, the final screen may resemble this: Thus, the given integral is approximate by 0.9

6 Question No. 6 of 10 10x 3 Question 6. Find the approximate maximum value correct to 3 decimals for the function y = x. + 1 Question #06 (A) (B) (C) (D) (E) The precision is wrong. The precision is wrong. Wrong precision The precision is incorrect. E. Correct! Employ the first or second derivative test. You may apply the first or second derivative test. However, most graphing aids are equipped with procedures for finding maxima. Employ the appropriate keystrokes to arrive at the answer quickly. The last screenshot may look like this Therefore, the maximum value correct to 3 decimals is 3.70.

7 Question No. 7 of 10 3 Question 7. Provide an approximation correct to 4 decimals of the real root of x x 3 = 0. (A) i (B) i Question #07 (C) (D) (E) This is not a real number. This is not a real number. The precision is wrong. The precision is incorrect. E. Correct! Graph the function to realize that the equation indeed has a real root (x-intercept). Use the appropriate procedure of your graphing aid to uncover an approximation of the solution correct to 4 decimals. Graph the left hand side of the equation using your graphing aid and use the appropriate keystrokes to approximate the x-intercept. The last screen shot may resemble this Therefore the zero of the given function correct to 4 decimals is

8 Question No. 8 of 10 Question 8. Find an approximation correct to 3 decimals of the zero of f ( x) = sin( x 1) x over [-1 0] Question #08 (A) (B) (C) (D) (E) A. Correct! Graph the function and approximate its zero using your graphing aid. This value falls outside of the interval [-1 0]. The precision of this answer is incorrect. This value is outside of the interval [-1 0]. E. Incorrect The precision of this answer is not correct. Use the appropriate keystrokes to graph and approximate the zero of the given function. Your last screenshot may look like this: Hence the zero correct to 3 decimals is

9 Question No. 9 of 10 Question 9. Find all the root of cos x = x correct to six decimal places. Question #09 (A) (B) (C) 0.64 (D) (E) The solution must be between 0 and 1 since the cosine of an angle is always between -1 and 1 and the right hand side is zero The solution must be between 0 and 1 since the cosine of an angle is always between -1 and 1 and the right hand side is zero The precision of this answer is not correct. D. Correct! Graph y = cos x x and approximate its x-intercept using your graphing utility. E. Incorrect The precision of this answer is wrong. Graph y = cos x x and approximate its x-intercept using your graphing utility. The last screenshot may look like this Thus, the zero correct to 6 decimals is

10 Question No. 10 of 10 Question 10. Find the approximate x-coordinate of the point of intersection of y= cos x and answer correct to two decimals. y = x. Provide your Question #10 (A) 1.4 (B) 1.39 (C) (D) (E) 0.64 The answer must be between 0 and 1 since cosine is always between -1 and 1 and the square root is positive. The answer must be between 0 and 1 since cosine is always between -1 and 1 and the square root is positive. The precision is incorrect. The precision is incorrect. E. Correct! Graph y = cos x x and approximate its x-intercept using your graphing utility. cos x = x in The two graphs intersect provided their y values coincide. Therefore we must solve the equation order to get the x-coordinate of the point of intersection. Use your calculator and appropriate keystrokes to approximate the x-intersect of y = cos x x. The last screenshot may resemble this Therefore the x-coordinate of the point of intersection correct to decimals is 0.64