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 𝑓(𝑥) 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝑓(𝑥) =
a. 2/√2
d. zero
b. √2/2
sin
𝑓𝑜𝑟 𝑥 < 4
3𝑥 + 2 𝑓𝑜𝑟 𝑥 > 4
c. √2/3
d. 2/7
.
d. 3/√2
4. Find the average rate of change of the function 𝑠(𝑥) = 𝑥√2𝑥 + 1 on the interval [4,12].
a. 3
b. 4
c. 5
d. 6
5. Find the instantaneous rate of change of 𝑝(𝑡) = √4𝑡 + 1 𝑒 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. 126.5𝑖𝑛
b. 125.5𝑖𝑛
c. 128.5𝑖𝑛
d.127.5𝑖𝑛
7. Find the radius of curvature at any point in the curve 𝑦 + ln cos 𝑥 = 0.
a. cos 𝑥
b. 1.5707
c. sec 𝑥
d. 1
8. Find the radius of curvature of a parabola 𝑦 − 4𝑥 = 0 at point (4, 4).
a. 22.36 𝑢𝑛𝑖𝑡𝑠
b.25.78 units
c. 20.33 𝑢𝑛𝑖𝑡𝑠
d. 15.42 units
9. Locate the point of inflection of the curve 𝑦 = 𝑓(𝑥) = 𝑥 𝑒 .
a. −2 ± √3
b. 2 ± √2
c. −2 ± √2
d. 2 ± √3
10. Find the critical points in the curve 2 + 12𝑥 − 𝑥 .
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 4𝑥 + 8𝑦 = 36.
a. 64𝑥
b. (-9/4)𝑦
c. 32xy
d. (-16/9)𝑦
12. Differentiate (𝑥 + 2) / .
a. [(𝑥 + 2) / ]/2
b. 𝑥/(𝑥 + 2)
/
c.2x/(𝑥 + 2)
13. What is the derivative with respect to x of (𝑥 + 1) − 𝑥 .
a. 3x + 6
b. 3x – 3
c. 6x – 3
/
d. (𝑥 + 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