MAT265 Review Problems for Exam 1 Domain of a Function (review from algebra) 1. Find the domain of the following functions. a. 9)(2 xxh b. xxexgx273)(2 c. 2315)(xxxf Properties of a Function (review from algebra) 2. Determine whether fis even, odd, or neither. a.  xxf cos)(  b. 24)(3 xxxg c. xxxh )( Algebra of Functions (review from algebra) 3. Suppose13)(  xxf, xxxg 2)(2, and xexh54)( . Find the following. a. )1)(( gf  b. )2)(( fg c. ))(( xgf  d. ))(( xfg  e. ))(( xhf  f. ))(( xfh Limit of a Function 4. Evaluate the function 992)(xxxf at the given numbers (correct to six decimal places). Use the results to guess the value of the limit, )(lim9xfx. x f(x) 8.9 8.99 8.999 9.1 9.01 9.001 limit5. Graph the function225211313)(2xifxxifxxifxxf Calculate the following. a. )(lim1xfx b. )(lim1xfx c. )(lim1xfx  d. )(lim2xfx e. )(lim2xfx f. )(lim2xfx 6. Graph the function 220cos302)(2xifxxifxxifxxg Calculate the following. a. )(lim0xgx b. )(lim0xgx c. )(lim0xgx d. )(lim2xgx e. )(lim2xgx f. )(lim2xgx Limit Laws 7. Given that3)(lim7xfxand9)(lim7xgx, evaluate the limit )()(3)(2lim7xfxgxxfx. 8. Algebraically calculate the following limits. a. 168103lim224xxxx b. 4573lim3xxx c. ttt9)3(lim20 d. xxx22lim2 Squeeze Theorem 9. Use the squeeze theorem to evaluate17sin)(lnlim21xxx. Continuity 10. Explain why each function is discontinuous at the given x value. a. 23225)(xifxxifxxf at x = 2 b. 14311)(2xifxxifxxf at x = -111. Find the values for c that will make the functionfcontinuous on  ,.14217)(2xifxcxifcxxf 12. Graph the function xxf )(, then determine the values of for whichfis discontinuous. Intermediate Value Theorem 13. Use the intermediate value theorem to show that the equation xx 213 has a solution in the interval  1,0. Limits involving infinity 14. Calculate the following limits, if they exist. a. 1367lim22xxx b. 741lim32xxx c. 629lim2xxx d. xxx319lim2 e. 7316lim2xxx f. xxx214lim2 15. Suppose 2ln25)(xxxf. Calculate the following limits. a. )(lim1xfx b. )(lim1xfx  c. )(lim1xfx Derivative, Tangent Line, Velocity 16. Find an equation of the tangent line to the curve at the given point. (Use the limit definition, i.e., either Definition 1 on page 75 or equation 2 on page 76.) a. 183)(2 xxxf at = 2 b. 2)(  xxf at = 6 c. xxf1)(  at = 1 17. The displacement (in meters) of a particle moving in a straight line is given by188)(2 tttswhere t is measured in seconds. a. Find the average velocity over the interval 4,3. b. Find the instantaneous velocity when t = 4.Derivative as a Function 18. Use the limit definition of the derivative to find)(' xf. a. 22)( xxxf  b. xxf311)( c. 53)(  xxf 19. Graph the given function and then determine the value(s) of x for whichfis not differentiable and explain why. a. 3153)(  xxf b. 21)(  xxf Basic Differentiation Formulas 20. Supposexxgxxxf cos2)(54sin3)(2 and1)0(' g. Find)0('f. 21. Supposexxxxxxgxf32cos4)()( and6)(' g. Find)('f. 22. Find an equation of the normal line to the parabola452 xxy that is parallel to the line5315  yx. 23. The position of a small rocket that is launched vertically upward is given by10405)(2 ttts for25.80  t, where t is measured in seconds and)(ts is measured in meters above the ground. a. Find the velocity of the rocket for any time t. b. Find the time at which the velocity is zero. c. Use part (b) to find the maximum height reached by the rocket.Note: There is a reasonable assumption that most of these answers are not incorrect. 1. a. ),3[]3,(  b. ),4()4,0[  c. ]5,2()2,(  2. a. even b. neither c. odd 3. a. 3 b. 0 c.1632 xx d. 31292 xx e. 1125xe f. 5154xe 4. 31)(lim9xfx 5. a. –2 b. –2 c. –2 d.7 e. 1 f. DNE (does not exist) 6. a. 2 b. 2 c. 2 d. 2 e. 4 f. DNE (does not exist) 7. 45 8. a. 47 b. 78 c. –6 d. DNE (does not exist) 9. 0 10. a. DNExfx)(lim2 b. )1()(lim1fxfx 11. c = 3 12. f is discontinuous at every integer. 13. define 12)(3 xxxf and let N=0, then apply Inter. Value. Thm. 14. a. –2 b. 0 c. 21 d.0 e. 31 f. 0 15. a.  b.  c.  (does not exist) 16. a. 114  xy b. 2141 xy c. 2 xy 17. a. –1 b. 0 18 . a. xxf 21)(  b. 2)31(3)(xxf c. 5323)(xxf19. a. f is not differentiable at 5x, b/c the tangent line to the graph of f at 5x is vertical. b. f is not differentiable at 1x, b/c f has a “corner” at 1xand f has no tangent at 1x. 20. 8 21. 2 22. 253565  xy 23. a. 4010)(  ttv b. 4t c. 90