differential calculus exam

DIFFERENTIAL CALCULUS
TECHNOLOGICAL UNIVERSITY OF THE PHILIPPINES
Midterm in Differential Calculus
Multiple Choice: Choose the best answer in the given choices. Write your answer on the given answer sheet. Use Capital
Letter. Write “E” if your answer is not among the choices.
3𝑥+2 −3 −3 −3
1. Evaluate lim a. 3 b. c. 3𝑥 2 d. 4
𝑥→−1 𝑥+2 𝑥 𝑥 𝑥
a. –1 b. 1 c. 2 d. -2
13. 𝑓(𝑥) = √𝑥 + 2, the derivative is _____
2. Evaluate lim𝜋(sin 𝑥 + cos 𝑥) 1 2
a. b.
𝑥→
2
√𝑥+2 √𝑥+2
1
√2 −√2 c. d. 2√𝑥 + 2
a. 1 b. -2 c. d. 2√𝑥+2
2 2

sin 3𝑥 14. If 𝑥 2 + 𝑦 2 + 3𝑥 − 2𝑦 − 25 = 0, then its derivative
3. Evaluate lim𝜋 E. 2/pi
𝑥→ 2 𝑥 is
2𝑥+3 2𝑥−3
a. 1 b. 3 c. 1/3 d. –1 a. b.
2+2𝑦 2+2𝑦
2𝑥+3 2𝑥−3
c. d.
𝑥 2 +3𝑥+9 2−2𝑦 2−2𝑦
4. Evaluate lim
𝑥→∞ 4−𝑥 2
a. –1 b. –1/3 c. -1/9 d. 4/9 15. Find y’, given that 𝑥𝑦 + 2𝑥 + 3𝑦 − 27 = 0
2+𝑦 3+𝑦
a. b.
ln 𝑥 𝑥−3 −𝑥−2
5. Evaluate lim l’hopital E. 1 2+𝑦 3−𝑦
𝑥→1 𝑥−1 c. d.
−𝑥−3 −𝑥−2
a. -1 b. LDNE c. 2 d. 0/0 16. If 𝑓(𝑥) = 𝑥 + 9𝑥 2 + 3𝑥 + 2, then 𝑓′′′(𝑥) is
3

______
𝑒 𝑥 −1
6. Evaluate lim a. 6x + 18 b. 6 c. 0 d. 3x
𝑥→1 𝑥−1
a. e b. 1 c. –1 d. LDNE
17. 𝑓(𝑥) = √3𝑥 + 2, then 𝑓′′(𝑥) is ______
−9 −18
𝑥 2 + 3, 𝑥 ≥ 1 a. b.
7. Given 𝑓(𝑥) = { , which of the 3/2
(3𝑥+2) 3/2 (3𝑥+2)
𝑥 − 2, 𝑥<1 −3 −9
following statement is true c. d.
4(3𝑥+2)3/2 4(3𝑥+2)3/2
I. f(1) exist
II. f is continuous at x=1 1
18. The third derivative of 𝑓(𝑥) = is ______
𝑥2
III. f has a jump discontinuity 24 −24 24 −24
a. I only b. I and II c. I and III d. all of them a. b. c. d. 4
𝑥5 𝑥5 𝑥4 𝑥


8. Given 𝑓(𝑥) = |𝑥| , which of the following statement 19. 𝑓(𝑥) = ln(𝑥 3 + 2𝑥 + 1), 𝑓′(𝑥) is ______
is true 3𝑥 2 3𝑥 2 +2
a. b.
𝑥 3 +2𝑥+1 𝑥 3 +2𝑥+1
a. f is continuous 2 3𝑥+2
b. f has a jump discontinuity c. d.
𝑥 3 +2𝑥+1 𝑥 3 +2𝑥+1
c. f has a infinite discontinuity
d. f has a remove discontinuity 20. 𝑓(𝑥) = log 5 (3𝑥 + 2), 𝑓′(𝑥) is equal to ____
3log5 𝑒 log5 𝑒
a. b.
3𝑥+2 3𝑥+2
9. For what value of k, does 3 1
c. d.
𝑥 + 𝑘, 𝑥 ≥ 3 (3𝑥+2) log5 𝑒 (3𝑥+2) log5 𝑒
𝑓(𝑥) = { 𝑥 21. The derivative of 𝑓(𝑥 ) = ln[ (𝑥 + 2)2 (𝑥 − 2)4 ] is ____
𝑥<3
𝑥−2 1 4 4 2
the function is continuous a. + b. +
𝑥+2 𝑥−2 𝑥+2 𝑥−2
1 1 2 4
a. 1 b. 1/3 c. –2 d. 0 c. + d. +
𝑥+2 𝑥−2 𝑥+2 𝑥−2

10. Which of the following is true about a greatest 22. Find y’ in the equation ln 𝑥𝑦 + 3𝑥 − 2𝑦 + 3 = 0
integer function. 3𝑥𝑦+𝑦 −3𝑥𝑦−𝑦
a. b.
a. It is continuous for every real number 𝑥−2𝑥𝑦 𝑥−2𝑥𝑦
3𝑥𝑦−𝑦 3𝑥𝑦+𝑦
b. It is continuous for every integer c. d.
𝑥−2𝑥𝑦 𝑥+2𝑥𝑦
c. It is not a function
d. It has a jump discontinuity 23. Given that ln(𝑥 + 𝑦) + 2𝑥 − 2𝑦 + 100 = 0, then y’
is equal to ____
11. Find the derivative of 𝑓(𝑥) = 𝑥 3 + 2𝑥 + 1 1+2𝑥+2𝑦 1−2𝑥+2𝑦
a. b.
a. 3𝑥 2 + 2 b. 3𝑥 + 3 2𝑥+2𝑦−1 2𝑥+2𝑦−1
1−2𝑥−2𝑦 1+2𝑥+2𝑦
c. 3𝑥 4 + 2𝑥 2 + 1 d. 5 c.
2𝑥+2𝑦−1
d.
−2𝑥−2𝑦+1