School of Engineering
Dept. of Mathematics
Course Name: Calculus and Differential Equations
Course Code: MAT2301
Academic Year 2025-26 ODD SEMESTER
Module 1:
Differential Calculus:
Introduction, Application, Limit, continuity, Rolle’s theorem, Lagrange’s Mean Value
Theorem, Indeterminate Forms (𝟎
𝟎, 𝟎 × ∞, ∞ −∞, 𝟏∞ ) and L'Hospital's rule; Partial
derivatives, total derivative, Taylor’s and Maclaurin (self-study) theorems, Euler’s
Theorem, Jacobians, Maxima, minima and saddle points; Method of Lagrange multipliers
(self-study).
Questions:
Limits:
1. Evaluate the limits
a. lim
𝑥→−1𝑥10+𝑥5+1
𝑥−1 b. lim
𝑥→0cos𝑥
𝜋−𝑥 c. lim
𝑥→0𝑎𝑥+𝑏
𝑐𝑥+1. d. lim
𝑥→3√𝑥+√3
𝑥+3 .
2. Evaluate the limits
a. lim
𝑥→2 3𝑥2−4𝑥+1
2𝑥2+5𝑥+3 b. lim
𝑥→3𝑥2−9
𝑥+2.
Continuity:
3. Discuss the continuity of 𝑓(𝑥)=𝑥+1
𝑥2+1 at 𝑥=1.
4. If 𝑓(𝑥)={𝑥2−1
𝑥+1 , 𝑥 ≠ −1
−2, 𝑥= −1. Test the continuity of the function at 𝑥=−1.
5. Test the continuity of the function defined as follows at 𝑥=1.
𝑓(𝑥)={3𝑥 − 5, 𝑥 ≠ 1
2, 𝑥= 1.
6. Show that 𝑓(𝑥)=|𝑥|, is continuous at 𝑥 = 0.
Rolle’s theorem:
7. Verify Rolle’s theorem for the function 𝑓(𝑥)=𝑥2+2 in [−2,2].
8. Verify Rolle’s theorem for the function 𝑓(𝑥)=𝑥2+2𝑥−8 in [−4,2].
9. Verify Rolle’s theorem for the function 𝑓(𝑥)=𝑥2−6𝑥+5 in [0,6].
10. Verify Rolle’s theorem for the function 𝑓(𝑥)=sin𝑥
𝑒𝑥 in [0,𝜋].
Lagrange’s mean value theorem:
11. Verify Lagrange’s mean value theorem for the function 𝑓(𝑥)=𝑥2 in [2,4].
12. Verify the mean value theorem for the function 𝑓(𝑥)=𝑥2−4𝑥 −3 in [1,4].
