Math 1a midterm 2 a salt crystal is growing in a

Question

MATH 1A MIDTERM 2-

Problem 1- (i) State Rolle's Theorem.

(ii) State the Mean Value Theorem.

(iii) Prove the Mean Value Theorem. You may assume Rolle's Theorem.

Problem 2- In each of the following cases, evaluate dy/dx.

(i) y = sin(x) ln |x|.

(ii) y = x²/cos x

(iii) y = (10x⁴ - 5)²⁰

(iv) y = sin(x sin x)

(v) y = tan(x²)e^{x^3}

(vi) y⁴x - 3y = e^2x (you should leave your answer in terms of y and x)

Problem 3- Showing your working carefully, calculate dy/dx when y = x^{e^x}. Give your answer in terms of just x.

Problem 4- I start with 5 ties to wear on special occasions. My hair-dresser tells me that my separation anxiety is because I don't have enough nice clothes. Taking the advice to heart, I start shopping and my tie collection starts to grow exponentially. After 3 days I have 12 ties. How long will it be before I have 100 ties? You do not need to simplify or evaluate your answer.

Problem 5- Which point on the graph of y = √x is closest to the point (4, 0)?

Problem 6- A salt crystal is growing in a super-saturated solution of salt. It is a perfect cube and its length, width and height are all growing at a rate of 1 mm per day. What is the rate of increase of the volume of the cube when its length, width and height all equal 10 mm?

Problem 7- Showing your working carefully, evaluate

lim_x→0(sin²(x)/4x²).

If you use a rule to change the limit, then you should name that rule each time you use it.

Problem 8- What is 9log₄₉7?