SAMPLE FINAL EXAM MATH 16A WINTER 2017

Transcription

1 SAMPLE FINAL EXAM MATH 16A WINTER 2017 The final eam consists of 5 parts, worth a total of 40 points. You are not allowe to use books, calculators, mobile phones or anything else besies your writing utensils. You shoul write the answer to a question in the empty space after the question. The last 3 pages of the printe eam can be use as scratch paper. Feel free to tear the last sheet if you wish. Writing only the answer to a question, even if correct, amounts to very few points, if any. Show your work unless eplicitly specifie otherwise. Please write your name clearly below. NAME: You have 2 hours to work on the 5 parts. Rea everything carefully. Take your time. Goo luck! 1

2 2 MATH 16A Part I: Easy limits an erivatives [14p] I.a. Compute the following limits using irect substitution. Just the answer will o for this question. lim 4 lim ( + 9) = 8 3 = lim 8 ( 1)( 2)( 3) = tan() + 1 lim = I.b. Compute quickly the following 10 erivatives. Be careful because the last two are higher orer erivatives. Just the answer will o for this question. If your answer is significantly more complicate than it nees to be, you will lose points. [33] = [ ] = [ 4] = [ ] = [sin + 2 cos ] = [ 5 4] = [sec()] = [sin(sin())] = 2 2 [cos(4)] = 10 [ ] =

3 SAMPLE FINAL EXAM 3 Part II: Harer limits an erivatives [6p] II.a. Compute the following erivative. Show your work. [ ] cos 2 + sin II.b. Compute the following limit. Show your work. Don t use L Hopital s rule lim

4 4 MATH 16A Part III: Analyzing a function [10p] In this part, you will have to analyze the following function: f() = You can use the empty space below to make the usual table with, f, f, f, which you will graually complete working out the questions below. Please use this empty space to raw the graph of f. (This will be the final sub-question.) (a) Fin the -intercepts an y-intercepts of the graph of f. Hint. The function f factorizes as follows: f() = (2 1) 2 ( + 1). (b) Compute f an f the first an secon erivative of f.

5 SAMPLE FINAL EXAM 5 (c) Fin the critical point(s) of f. Classify them: are they relative minima, relative maima, or neither? On which intervals is f increasing? On which intervals is f ecreasing? Note. When you plug in, you will have to compute some rather ugly fractions. Compute them carefully. Leave the answer in fraction form. () Fin the inflection point(s) of f. Show your work, on t just write the answer. (e) Use the results of parts (a) () above to raw the graph of the function f. Preferably, use the empty space on the previous page.

6 6 MATH 16A Part IV: Optimization problem [6p] An architect has to buil a house with 3 rooms, respecting the layout in the picture below. Two of the rooms are equal an rectangular an the thir one is square-shape. y y y Because of limite construction materials, the total length of the walls (both interior an eterior) cannot ecee 264 feet. Given this constraint, what is the maimal possible area of the house an what is the shape for which this maimum is attaine?

7 SAMPLE FINAL EXAM 7 Part V: True or false [4p] For each of the following, state whether it is true or false. No justification require for parts (a) an (b). For part (c), justification is require. (a) [1p] If a function is everywhere continuous, then it is everywhere ifferentiable. (b) [1p] The function f() = sin is not ifferentiable at infinitely many -values. (c) [2p] If the function f is everywhere (efine an) continuous an has local etrema at eactly 2017 ifferent -values, then f amits at least one absolute etremum. If yes, eplain why. If no, eplain why not. Note. To keep things clear, f is not allowe to be constant on any open interval.

8 8 MATH 16A (scratch paper)

9 SAMPLE FINAL EXAM 9 (scratch paper)

10 10 MATH 16A (scratch paper)