Math 1431 Final Exam Review

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1 Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back and re-schedule. Best times to find one is 9 p.m. midnight. Do the Teacher Evaluations on CASA, +1 point on the final exam grade. Practice Final Exam is for practice and extra credit; 5% of the highest score is added to the final exam grade. The final exam raw score will replace one missed test or lowest test grade (if it s better). Math 1431 Final Exam Review 1

2 1. Find the following limits (if they exist): a. lim x ( x 1) x5 b. lim 1 x 1 x1 c. x 1 lim x1 x 1 d. lim x x 1 4 x e. 1 1 x x x lim 0 ln( 1) Math 1431 Final Exam Review

3 Recall: sin( ax) a lim x 0 bx and b 1 cos( ax) lim 0 x0 bx f. 4x lim x0 sin(10 x ) g. 1 cosx lim x0 sin6x. Use the following graph to find the following limits, if they exist. a. lim f x x 1 b. lim f x x 1 c. lim f x x 1 What type of discontinuity occurs at x = 1? at x = 0? at x = -3? Math 1431 Final Exam Review 3

4 3. Describe any discontinuities of the given function. 5 x 3 x 9 f ( x) 3 x 3 x 3 10 x 3 Step 1: Is the function defined at the x-value? Step : The limit must exist. Step 3: lim f( x) = f(a)? i.e. Compare #1 and # above. x a Hence, at x = f(x) is: CONTINUOUS or DISCONTINUOUS Type: JUMP REMOVABLE INFINITE Hence, at x = f(x) is: CONTINUOUS or DISCONTINUOUS Type: JUMP REMOVABLE INFINITE Math 1431 Final Exam Review 4

5 5. Find values of A and B so that the function is continuous: Find f(4). 3x 1 x4 f x A x 4 Bx 1 x 4 Find each one-sided limit: lim f( x) x 4 and lim f( x) x 4 These must be equal! Math 1431 Final Exam Review 5

6 Know the definition of the derivative. f ' x f f lim x h x h 0 h, provided the limit exists. 6. Use the limit definition of the derivative to find f '( x ) for f ( x) x 1. Then find f '(1). 7. Find the derivative of the following: a. f x 8cot x Then find f'. Math 1431 Final Exam Review 6

7 b. f x sin x sinh( x) 3 f x 3x tan x arctan( x) 4arcsin(4 x) c. d. f x x 1 3 ( x 1) Then find df. dx x 0 Math 1431 Final Exam Review 7

8 f x 4 cosh ( x) ln( x 3 x) e. x e f. f ( x) (log x) x Know how to find a second derivative of all types of functions. Math 1431 Final Exam Review 8

9 8. Given y lines at that point. xy 6. Compute dy dx (6,4) and give the equations of the tangent and normal 9. Find f 1 Recall: f given f 1, f (3) 1, f ' 1, f ' 3, f ' '( b) f '( a) 1 1 Math 1431 Final Exam Review 9

10 10. Suppose we are given the data in the table about the functions f and g and their derivatives. Find the following values. x f (x) f (x) g (x) g (x) a. h' 1 if h x e gx b. h' if h x f x g x Math 1431 Final Exam Review 10

11 11. Use the intermediate value theorem to show that the function the interval 1,. f x 3 ( ) x 1 has a root on 1. Verify that the Rolle s Theorem applies to the function f x cosx over find all points in this interval that satisfy Rolle s Theorem. If f continuous on 0,? If f differentiable on 0,? f(0) = f( ) = 0,. Then a,b. If f a f b 0, then there is at least one number c in Recall: Suppose that f is continuous on the closed interval open interval which f ' c 0. Make sure to review the mean value theorem (Section 3.). a,b and differentiable on the a,b for Math 1431 Final Exam Review 11

12 13. Use the guide to curve sketching to sketch Domain of f( x ): x-intercept(s): f ( x) x 4 3 x 4. Asymptotes: y-intercept(s): f '( x) 4x 1x 3 Find when '( ) 0 f x : Find when f '( x) is undefined: Critical Points: f( x ): f '( x ) : f ''( x) 1x 4x Find when f ''( x) 0: Find when f ''( x) is undefined: f( x ): f ''( x ) : Math 1431 Final Exam Review 1

13 14. The following graph is the graph of f ''( x ) of a polynomial function f. Determine where f is concave up and concave down. a. f is concave up on: b. f is concave down on: 15. The value x = a is a critical number for f (x). Use the second derivative test to classify it as a local maximum, local minimum or neither. 3 f x x 1 x, x 8 Math 1431 Final Exam Review 13

14 16. Find the absolute extremum of the function f ( x) x 4 e on [-, ]. 17. Use differentials to approximate 63. f x h f x f ' x h Recall: Math 1431 Final Exam Review 14

15 d dx x Recall: F' x f t dt f x and using the FTOC f x dx F b F a a F is the antiderivative. d dx x 1 dt. 18. Evaluate 3t 4 b a, where f x 19. Compute the upper sums for the function partition P = {0, 1, }. x on the interval [0, ] associated with the Math 1431 Final Exam Review 15

16 0. Given the graph of f (x) below with the area of region A equal to 7/3, region B is 34/3 and region C is 7/3. a. Find the area of the region bounded by f (x) and the x-axis between and 4. 4 b. Find f x dx. 1. Integrate: 3 3 a x x x dx b. x x 3 dx x 3 Math 1431 Final Exam Review 16

17 c. 1 4 x x dx d. 0 3 cos x dx e. 11xe x dx Math 1431 Final Exam Review 17

18 f. 5sinh 10x dx g. csc cot x x dx h. sin 5 xcos x dx Math 1431 Final Exam Review 18

19 i. x x 3 4 dx j. 0 ln sinh( x) cosh( x) 1 dx Math 1431 Final Exam Review 19

20 k. 1 x x 1 dx l. 4 x dx. Suppose the distance, in feet, covered by a car moving along a straight road t seconds after starting from rest is given by a. the position at t = 5. f ( t) t 48t. Find: b. the velocity at t = 7. c. the acceleration at t = 7. Math 1431 Final Exam Review 0

21 3. A ball is thrown straight up with an initial velocity of 18 ft/sec, so that its height (in feet) after t seconds is given by f ( t) 16t 18t and is the ball rising or falling at this time?. What is the instantaneous velocity at time t = 5, 4. Suppose a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How fast is the radius increasing when the radius is 5 centimeters? Picture: Given: Question: Relation: Review all Related Rates problems (Section 3.1) from the class notes, quizzes, practice tests! Math 1431 Final Exam Review 1

22 5. Find two numbers whose sum is 10 and the sum of their squares is a minimum. Review all Optimization Problems (Section 5.1) from the class notes, quizzes, practice tests! Math 1431 Final Exam Review

23 Recall: Removable Discontinuity occurs when: lim f ( x ) f(c) xc Jump Discontinuity occurs when: lim f( x) and lim f( x) exists, but are not equal xc xc Infinite Discontinuity occurs when: lim f( x) on at least one side of c. Infinite discontinuities are generally associated xc with having a vertical asymptote. Determine if the following function is continuous or discontinuous at x = - and x = 3. x, x, 3 x x6 f( x) 1, x, 3 5 Math 1431 Final Exam Review 3

24 Find all points of discontinuity and classify them for: x 1, x0 f ( x) x 1, 0 x 1, x x 6 Math 1431 Final Exam Review 4

Calculus I Exam 1 Review Fall 2016

Calculus I Exam 1 Review Fall 2016 Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function