math calculus
MATH 005A Exam #2:
Instructions: Same instructions as last time try it first with no notes or books, then go back and make corrections. Do your work in separate columns. DO NOT attempt to do your work in the spaces provided!!!!
1. (20 pts) For the function:
𝑓 𝑥 𝑥 3−
𝑥 3 - here are the first and second derivatives:
𝑓′ 𝑥 3𝑥 3𝑥 𝑥
𝑓 𝑥 3 3𝑥 𝑥 𝑥− 𝑥
USE THESE FACTS Don t try to verify them. A) Find lim
𝑥→ 𝑓 𝑥 , lim
𝑥→− 𝑓 𝑥
B) Find all the critical numbers for f(x), and
C) Classify each of your answers from (B) as a local maximum, local minimum, absolute maximum, absolute minimum, or none of these.
D) On which intervals is f(x) increasing? Decreasing?
E) What is the range of f(x)?
F) Find all inflection points.
G) On which intervals is f(x) concave down? Concave up?
2. (6 pts) If y = [x + (x + cos3(x))3]4 find dy/dx. No need to simplify your work!
3. (6 pts) If g(x) = 3
si 3 , find g N need to simplify your work!
4. (8 pts) Find an equation for the tangent line to f(x) = sec 𝑥 at x = 1
5. (6 pts) State and prove the product rule
6. (12 pts) For the rotated hyperbola: x2 + 2xy 4y2 = 20: A) Show that it has NO horizontal tangents. B) Find the x- and y- coordinates of the place(s) where it has vertical tangents.
7. (10 pts) One car is six miles north of an intersection, heading south at 40 mph. Another car is 3 miles east of the same intersection, heading east at 50 mph. At what rate is the distance between the vehicles changing? Are they getting closer together, or farther apart?
8. (12 pts) P(x) = x3 - 5x + 2 on [0, 2] A) P(x) has a zero between what two consecutive integers? How do you know? B) Find the point in the interval [0, 2] that satisfies the conclusion of the Mean Value Theorem for P(x).
C) Find the maximum and minimum values for P(x) on [0, 2] and the place(s) where they occur.
9. (10 pts) You need to make a box with a square base and open top. The volume of the box must be 140 cm3. The base material costs $0.20 per cm2, and the side material costs $0.10 per cm2. Find the dimensions that minimize the cost of the material.
10. (10 pts) Find each of the following derivatives no need to simplify:
A) 𝑥
𝑥3 𝑓 𝑥
B) 𝑥
𝑓 𝑥 3