PROBLEM SET NO. 5_CALCULUS 1 September 25, 2023 Multiple Choice. Choose the correct letter with the correct answer to the following questions. 1. Evaluate the limit, lim → a. undefined 2. Evaluate the limit, lim a. -7/2 → b. 3/5 | | c. infinity b. -2/7 c. 7/2 3. Evaluate the limit, lim ๐(๐ฅ) ๐๐๐ฃ๐๐ ๐กโ๐๐ก ๐(๐ฅ) = a. 2/√2 d. zero b. √2/2 sin ๐๐๐ ๐ฅ < 4 3๐ฅ + 2 ๐๐๐ ๐ฅ > 4 c. √2/3 d. 2/7 . d. 3/√2 4. Find the average rate of change of the function ๐ (๐ฅ) = ๐ฅ√2๐ฅ + 1 on the interval [4,12]. a. 3 b. 4 c. 5 d. 6 5. Find the instantaneous rate of change of ๐(๐ก) = √4๐ก + 1 ๐ when t = 0. a. 3 b. 4 c. 5 d.6 6. Find the minimum area of tin sheet needed to make a closed cylinder having a volume of 108 in3? a. 126.5๐๐ b. 125.5๐๐ c. 128.5๐๐ d.127.5๐๐ 7. Find the radius of curvature at any point in the curve ๐ฆ + ln cos ๐ฅ = 0. a. cos ๐ฅ b. 1.5707 c. sec ๐ฅ d. 1 8. Find the radius of curvature of a parabola ๐ฆ − 4๐ฅ = 0 at point (4, 4). a. 22.36 ๐ข๐๐๐ก๐ b.25.78 units c. 20.33 ๐ข๐๐๐ก๐ d. 15.42 units 9. Locate the point of inflection of the curve ๐ฆ = ๐(๐ฅ) = ๐ฅ ๐ . a. −2 ± √3 b. 2 ± √2 c. −2 ± √2 d. 2 ± √3 10. Find the critical points in the curve 2 + 12๐ฅ − ๐ฅ . a. (2,18) & (−2, −14) b. (2,18) & (2, −14) c. (−2,18) & (2, −14) d. (−2,18) & (−2,14) 11. Find the second derivative of y by implicit differentiation from the equation 4๐ฅ + 8๐ฆ = 36. a. 64๐ฅ b. (-9/4)๐ฆ c. 32xy d. (-16/9)๐ฆ 12. Differentiate (๐ฅ + 2) / . a. [(๐ฅ + 2) / ]/2 b. ๐ฅ/(๐ฅ + 2) / c.2x/(๐ฅ + 2) 13. What is the derivative with respect to x of (๐ฅ + 1) − ๐ฅ . a. 3x + 6 b. 3x – 3 c. 6x – 3 / d. (๐ฅ + 2) / d. 6x + 3 14. A box is to be constructed from a piece of cardboard 20in2 by cutting equal squares from each corner and turning up the cardboard to form the side. What is the volume of the largest box that can be constructed? a. 599.95 in3 b. 592.59 in3 c. 579.90 in3 d. 622.49 in3 15. A banner is to contain 300cm2 of printed matter with margins of 5cm at each side and 10cm at the top and bottom. Find the overall dimension if the total area of the banner is minimum. a. 28.23cm, 48.8cm b. 22.24cm, 44.5cm c. 20.45 cm, 35.6cm d. 25.55cm, 46.7cm 16. A rectangular glass window is surmounted by a semicircle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter? a. 1 b. ½ c. 2 d. 2/3 17. The cost of fuel in running a locomotive is proportional to the square of the speed and is $25 per hour for a speed of 25 miles per hour. Other costs amount to $100 per hour regardless of the speed. What is the speed of which will make the cost per mile a minimum? a. 40 b. 55 c. 50 d. 45 18. Differentiate y=ex cosx2 a. –ex sinx2 b. ex (cosx2-2x sinx2) c. ex cosx2-2x sinx2 d. -2xex sin x 19. What is the derivative with respect to x of (x+1)3 – x3? a. 3x+6 b. 3x-3 c. 6x-3 d. 6x+3 20. Differentiate y = log10 (x2+1)2 a. 4x (x2+1) b. 4xlog10e/x2+1 c. log e(x) (x2+1)0 d. 2x (x2+1) 21. If y = (t2 +2)2 and t = x(1/2), determine dy/dx. a. 3/2 b. (2x2+2x)/3 c. 2(x+2) d. x(5/2) + x(1/2) 22. Find y’ if y = arc sin cos x a. -1 c. 1 d. 2 c. 3x+10 d. 3x2-5x c. 3/5 d. 4/5 c. -0.25 d. -0.875 b. -2 23. Find the second derivative of x3-5x2+x=0. a. 10x-5 b. 6x-10 24. Evaluate lim a. 1/5 → . b. 2/5 25. Find the second derivative of y=x-2 at x=2. a. 96 b. 0.375 26. Differentiate y=sec (x2+2) a. 2x cos (x2+2) b. –cos (x2+2) cot (x2+2) c. 2x sec (x2+1) tan (x2+2) d. cos (x2+2) 27. Evaluate the limit (ln x)/x as x approaches positive infinity. a. 1 b. 0 c. e d. infinity 28. If y = x(ln x), find d2y/dx2. a. 1/x2 d. -1/x2 b. -1/x c. 1/x 29. A point on the curve where the second derivative of a function is equal to zero is called a. maxima b. minima c. point of inflection d. point of intersection 30. The point on the curve where the first derivative of a function is zero and the second derivative is positive is called a. maxima b. minima c. point of inflection d. point of intersection