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Math 105, Midterm exam (2) Almajma’ah University Math 105 Engineering College 5/2/1431 Name: Level 2: Academic Number: Time allowed: 90 min. Midterm Exam (2) Question 1: Find: a. 𝑙𝑖𝑚 𝑥 2 +𝑥+7 𝑥→∞ 2𝑥 2 +3𝑥+4 3𝑥+𝑠𝑖𝑛3𝑥 b. 𝑙𝑖𝑚 𝑥→0 𝑠𝑖𝑛 𝑥+𝑡𝑎𝑛 4𝑥 Question 2: Find a and b so that the following function will be defined at all points 𝑥+1 𝑓𝑜𝑟 𝑥 < 1 𝑓𝑜𝑟 1 ≤ 𝑥 < 2 c. 𝑓(𝑥) = {𝑎𝑥 + 𝑏 3𝑥 𝑓𝑜𝑟 𝑥 ≥ 2 Question 3: See if the following function is continuous or not, if not mention the type of discontinuity. 1 ; f ( x) 1 ; c ; x 2 x2 x2 Question 4: Find the instant rate of change for the area of a circle with respect to its radius. Then find this rate when the radius equal to 3 1 Math 105, Midterm exam (2) Question 5: Find 𝒅𝒚 𝒅𝒙 d. 𝑥 3 − 2𝑥𝑦 + 𝑦 2 = 6𝑦 e. 𝑦 = 𝑥 3 √𝑥 2 + 𝑥 + 1 f. 𝑦 = (𝑥 3 + 𝑡𝑎𝑛 𝑥)6 Question 6: a) If 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 1 Find all the real number c that satisfy “Rolle’s Theorem” in the interval [0,4]. b) Use the definition of the first derivatives to find the first derivative of: f(x) = x2 + 2 Best Wishes Dr. SaMeH 2