Math 105B Final Exam: Solving Limits, Derivatives, Integrals, and Geometry Problems, Exams of Calculus

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Math 105B Final Exam

• There are 8 questions

• You have 2 hours

Name:

1. (3 marks) Evaluate the following limits without a calculator. a. limx→− 2 2 x^3 +4x x^2 +2−^8 x−^16

b. limx→ 0 sin√x^ x

c. limx→∞ 2 xx^22 +3− 5

3. (3 marks) Find antiderivatives for the following functions. a. f (x) = x^5 + sec^2 x + sin(2x) + ln 2

b. f (x) = 2 √^1 x + (^) 1+^1 x 2 + ex^ cos x + ex^ sin x

c. f (x) = (^) (3+cossin^ x x) 2

4. (3 marks) Evaluate the following integrals. a. ∫^0 π sin x + exdx

b. ∫^14 5 x + x^32 dx

c. ∫^01 x (^22) +1x dx

6. (6 marks) Airline A only allows carry on bags whose longest side plus its girth (girth=distancearound the box NOT using the longest side) does not exceed 112 inches. What are the di- mensions of the bag with two square ends that has largest volume?

7. (6 marks) A 5 foot 4 inch woman is walking towards a 12 foot tall streetlight at a rate of2 feet per second. When she is 12 feet from the base of the light, at what rate is the length of her shadow changing?