School of Chemistry
University of the Punjab, Lahore
BS Chemistry
1st semester
Paper: Mathematics (Final)
Instructor: H M Athar Farid
Attempt all questions.
8*5=40
1. State and prove the mean value theorem and also explain its geometrical significance.
2. In the following part a) and b).
i. Find the intervals on which the function is increasing and decreasing.
ii. Then identify the function’s local extreme values, if any, saying where they are taken
on.
iii. Which, if any, of the extreme values are absolute?
a) 𝑲(𝒕) = 𝟏𝟓𝒕𝟑 − 𝒕𝟓
b) 𝒀(𝒕) =
𝒕𝟑 −𝟑
𝒕−𝟐
,𝒕 ≠ 𝟐
3. Evaluate
𝟏
𝟏
a) ∫ 𝒕𝟐 𝐜𝐨𝐬 ( 𝒕 − 𝟏)
b) Find the area of the region enclosed by the parabola 𝒚 = 𝟐 − 𝒙𝟐 and the line 𝒚 =
−𝒙.
4. Evaluate
𝟐𝝅
a) ∫𝟎
𝐜𝐨𝐬 𝒛
√𝟒+𝐬𝐢𝐧 𝒛
𝝅
b) 𝒀(𝒕) ∫𝟎 𝐬𝐢𝐧𝟐 𝟓𝒓 𝒅𝒓
5. Find the total areas of the shaded regions