# Warm-up for Differential Calculus

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1 Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte: The first dy of school How mny of the problems should I do? ALL OF THEM How should I orgnize my work? You should show ll work in seprte sheets of looselef pper. If problem requires grph, then you should use grph pper. Keep your mterils in folder with 3-prongs. How will my techer know tht I ve done the work? Your techer will collect your notebook on the first dy of school. Your techer my choose to QUIZ or TEST you on this mteril if he or she feels it is necessry BE PREPARED! How well should I know this mteril when I return? You should recognize tht you ve seen this mteril before, nd you should lso be ble to nswer questions like the ones in this pcket. If the mteril is revisited in your next clss, it will only be for brief mount of time your techer will ssume tht ll you need is quick refresher. Note from your techers: We feel tht this summer work will truly help you succeed this yer. We understnd tht summer is time for relxtion nd fun, but it is impertive tht you spend some time before you return reviewing your mterils. This pcket is mndtory, nd you must tret it s you would ny other extremely importnt homework ssignment. You will be held ccountble for this mteril. We lso highly suggest tht you do bit of it t time in the weeks leding up to school don t leve it for the lst dy!!!

3 Wrm-up for Differentil Clculus pge 1 1. Grph f( x) ( x1) 4 using trnsformtions (shifting, stretching, reflecting, etc.) from the grph y x. Stte the trnsformtions needed.. For f( x) 3x 1x 4: i. Determine if the function hs mximum or minim nd explin how you know. ii. Algebriclly, determine the vertex of the grph. iii. Determine the xis of symmetry of the grph. iv. Algebriclly, determine the intercepts of the grph. v. Using the informtion in prts -d, grph the function by hnd. Plot dditionl points s needed in order to crete n ccurte grph. 3. Solve 3x x1 0 over the complex numbers. 4. Given the polynomil function Px ( ) ( x1) ( x ), do the following:. Find the x nd y intercepts of the grph. b. Determine whether the grph crosses or touches the x-xis t ech x-intercept. c. Describe the end behvior of the grph, tht is, describe wht hppens to the y-vlues s x increses. d. Using the informtion in prts -c, mke n ccurte sketch of the grph For the polynomil function g( x) x 5x 8x 15:. Determine the mximum number of rel zeros tht the function my hve. b. Given tht 3 is zero, determine the rel zeros of g. Fctor g(x) over the rel numbers. 6. Give the eqution of polynomil function with rel coefficients of degree 3 tht hs zeros of 5, 3 4i. 7. Sketch the grphs of these functions. Determine the mplitude, period, nd ny horizontl or verticl shifts. USE RADIANS. 1. y 43sin b. y 3cos 4

4 Wrm-up for Differentil Clculus pge 8. Give the eqution of sinusoid (sine or cosine function) tht meets the conditions:. hs mximum t (0,1) nd minimum t (5,-5) b. hs n mplitude of 3, period of 180, nd psses through the point (5,) 9. Given, 34 nd 1, find the following nd stte its domin. f. ( f g)( x) b. ( x) g c. f g( x) 10. Given the grphs of nd, find ( f g)( 1), ( fg )(5), nd ( g f)(4). 11. Let hx ( ) 6 x 1 Given hx ( ) ( f g)( x), nme nd. 1. Find the unknown vlue without using clcultor.. 3 x 343 b. logb 16 c. log5 x 4

5 Wrm-up for Differentil Clculus pge Solve ech eqution. Give n exct nswer when possible. When using clcultor, round your nswer to three deciml plces.. x 5 15 c. 8 e x 4 b. log x 9 d. log x 3 log x 6 e. x3 x 7 f. x x log 4 log Given log nd log , find ech of the following:. log 6 b. log 3 c. log 3 d. log Use trnsformtions to grph ech function: identify the bse function, then describe ny shifts, stretches, etc. Sketch n ccurte grph of the function. Determine the domin nd give the eqution of ny symptotes.. f x x1 ( ) 4 b. g( x) 1log x A 50 mg smple of rdioctive substnce decys to 34 mg fter 30 dys.. How much of the substnce is left fter 10 dys? b. How long will it tke for there to be mg remining? 17. Find the inverse of f ( x). 3x 18. If the point (3,-5) is on the grph of the function gx, ( ) wht point must be on the grph of g 1 ( x), the inverse function?

6 Wrm-up for Differentil Clculus pge Ech of the following grphs psses through the point P (1, ). Mtch the correct function to ech grph. Identify chrcteristic tht enbles you to determine which grph.. y x b. y sin(30 x) 1 c. y log x d. e. 1 3 y x y x x 4 5 x 1 0. Find the domin of f( x), x 11x10 gx ( ) 1 nd g f ( x). x 1. Simplify: x 8x16 3x 4. Multiply: x x6 3x9 x4 3. Simplify: 1 1 x y x y

7 Wrm-up for Differentil Clculus pge 5 4. For ech ngle on the unit circle shown, fill in the ngles (in degrees nd rdins). You my write your work on this sheet. Be sure to turn it in with your other work.

8 Wrm-up for Differentil Clculus pge 6 5. Give the exct vlue of ech of the following. 3. sin b. cos 3 c. sec(45 ) 7 d. cot(15 ) e. tn 6 6. Find ll vlues of on [0, ) for which the following re true. 3. cos b. sin c. cot 0 7. Verify ech identity. Remember to work only on one side of the eqution.. sin tn cos sec b. tn cot 1cos tn cot 8. Solve ech eqution on the intervl 0. 4sin 3 0 c. sin b. cos cos 1 d. tn 3 0

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