201-103-RE - Calculus 1 WORKSHEET: LIMITS

1 201-103-RE - Calculus 1 WORKSHEET: LIMITS1. Use the graph of the functionf(x) to answer each , orDNEwhere appropriate.(a)f(0) =(b)f(2) =(c)f(3) =(d)limx 0 f(x) =(e)limx 0f(x) =(f)limx 3+f(x) =(g)limx 3f(x) =(h)limx f(x) =2. Use the graph of the functionf(x) to answer each , orDNEwhere appropriate.(a)f(0) =(b)f(2) =(c)f(3) =(d)limx 1f(x) =(e)limx 0f(x) =(f)limx 2+f(x) =(g)limx f(x) =3. Evaluate each limit using algebraic , orDNEwhere appropriate.(a)limx 0x2 25x2 4x 5(b)limx 5x2 25x2 4x 5(c)limx 17x2 4x 33x2 4x+ 1(d)limx 2x4+ 5x3+ 6x2x2(x+ 1) 4(x+ 1)(e)limx 3|x+ 1|+3x(f)limx 3 x+ 1 2x2 9(g)limx 3 x2+ 7 3x+ 3(h)limx 2x2+ 2x 8 x2+ 5 (x+ 1)(i)limy 5(2y2+ 2y+ 46y 3)1/3(j)limx 04 2 cos(x) 5(k)limx 013 +x 13 xx(l)limx 62x+ 8x2 12 1xx+ 6(m)limx x2 2 x2+ 1(n)limx x 2 x(o)limx 76 2x 14(p)limx 1 3 3x(q)limx x4 104x3+x(r)limx 3 x 35 x(s)limx 3x3+x2 2x2+x 2x3+ 1(t)limx x+ 52x2+ 1(u)limx cos(x5+ 1x6+x5+ 100)(v)limx 22xx2 4(w)limx 13xx2+ 2x+ 1(x)limx 1x2 25x2 4x 5(y)limx 3 x2 5 + 2x 3(z)limx 02x+ sin(x)x4(A)limx 1 1x 1+ex2(B)limx 2x2 3x(C)limx 0 x+ 2 2 xx(D)limx 0+ex1 + ln(x)(E)

2 Limx x2+ 1 2x(F)limx 13 x 1 x 14. Find the following LIMITS involving absolute values.(a) limx 1x2 1|x 1|(b) limx 21|x+ 2|+x2(c) limx 3 x2|x 3|x 35. Find the value of the parameterkto make the following limit exist and be is then the value of the limit?limx 5x2+kx 20x 56. Answer the following questions for the piecewise defined functionf(x) described onthe right hand side.(a)f(1) =(b)limx 0f(x) =(c)limx 1f(x) =f(x) ={sin( x)forx <1,2x2forx > Answer the following questions for the piecewise defined functionf(t) described onthe right hand side.(a)f( 3/2) =(b)f(2) =(c)f(3/2) =(d)limt 2f(t) =(e)limt 1+f(t) =(f)limt 2f(t) =(g)limt 0f(t) =f(t) = t2fort < 2t+ 6t2 tfor 1< t <23t 2fort 2 ANSWERS:1.}

3 (a) DNE (b) 0 (c) 3 (d) (e) DNE (f) 2 (g) DNE (h) 12. (a) 0 (b) DNE (c) 0 (d) DNE (e) 0 (f) (g) 13.(a) 5(b)53(c) 5(d) 1(e) 1(f)124(g)16(h) 18(i)43(j) DNE(k) 29(l)136(m) 0(n) DNE(o) DNE(p) 0(q) (r) 1(s) 32(t) 0(u) 1(v) DNE(w) (x) DNE(y) DNE(z) (A) (B) (C)1 2(D) 0(E) (F)234. (a) DNE (b) (c) 1, limit is then equal to 96. (a) DNE (b) 0 (c) DNE7. (a) DNE (b) 4 (c) 10 (d) DNE (e)52(f) 4 (g) DNE8. (a) 0 (b) 0 (c)53 Name Pre- Calculus Rational functions worksheet For each of the rational functions find: a. domain b. holes c. vertical asymptotes d. horizontal asymptotes e. y-intercept f.

4 X-intercepts 1. 2226xxfxxx 2. 2221xfxx 3. 32fxx 4. 21xfxx 5. 22129xxfxx 6. 243xfxx 7. 21xxfxx 8. 221xxfxx 9. 2132xfxxx 10. 22923xfxxx 201-103-RE - Calculus 1 WORKSHEET: CONTINUITY1. For each graph, determine where the function isdiscontinuous. Justify for eachpoint by: (i) saying which condition fails in the definition of continuity, and (ii) bymentioning which type of discontinuity it is.(a)(b)2. For each function, determine the interval(s) of continuity.(a)f(x) =x2+ex(b)f(x) =3x+ 12x2 3x 2(c)f(x) =4 5 x(d)*f(x) =24 x2+1 x2 x 123.

5 For each piecewise defined function, determine wheref(x) is continuous (or where itis discontinuous). Justify your answer in detail.(a)f(x) ={2x 3x2forx 1log10(x) +xforx >1(b)f(x) = 2x3 xforx 0x2 3xfor 0< x <2x2 8xforx >24. Find all the value(s) of the parameterc(if possible), to make the given functioncontinuous everywhere.(a)f(x) ={c 3x x2+ 2cforx 02x5+c(x+ 1) + 16forx >0(b)f(x) ={2(cx)3+x 1forx 12cx+ (x 1)2forx >1(c)f(x) = 3x+cforx < 1x2 cfor 1 x 23forx >25.*Consider the functionf(x) =bxc, the greatest integer function (also called the floorfunction or the step function). Where is this function discontinuous?6.*Find an example of a function such that the limit exists at everyx, but that hasan infinite number of discontinuities.}}}

6 (You can describe the function and/or write aformula down and/or draw a graph.)PARTIAL ANSWERS:1. (a)x= 0,3 (b)x= 2,0,12. (a)R(b)R\{ 1/2,2}(c) ( ,5] (d) ( 3,2) ( 2,2) (2,4)3. (a) discontinuous only atx= 1 (b) discontinuous only atx= 24. (a)c= 8 (b)c= 1,0,1 (c) no solution possible5. discontinuous at every integer,x=.. , 3, 2, 1,0,1,2,3, ..6. many answers are possible, show me your solution! 201-103-RE - Calculus 1 WORKSHEET: DEFINITION OF THE DERIVATIVE1. For each function given below, calculate thederivative at a pointf (a)using the limit definition.(a)f(x) = 2x2 3xf (0) =?(b)f(x) = 2x+ 1f (4) =?(c)f(x) =1x 2f (3) =?2. For each functionf(x) given below, find thegeneral derivativef (x)as a new function by using the limit definition.)

7 (a)f(x) = x 4f (x) =?(b)f(x) = x3f (x) =?(c)f(x) =xx+ 1f (x) =?(d)f(x) =1 xf (x) =?3. For each functionf(x) given below, find theequation of the tangent lineat the indicated point.(a)f(x) =x x2at (2, 2)(b)f(x) = 1 3x2at (0,1)(c)f(x) =12xatx= 1(d)f(x) =x+ xatx= 1 ANSWERS:1. (a)f (0) = 3 (b)f (4) = 1/3 (c)f (3) = 12. (a)f (x) =12 x 4(b)f (x) = 3x2(c)f (x) =1(x+1)2(d)f (x) = 12x3/23. (a)y= 3x+ 4 (b)y= 1 (c)y= 12x+ 1 (d)y=32x+12 Derivative Practice Worksheet Name: _____ Solve the derivatives for using basic differentiation. 1. y = 3 2. 24g xx 3. 2236h ttt 4. 324s ttt 5. 23124xxfxx 6. 5yx 7. 4372135g xxxx 8.

8 212f xxx 9. 325yx 10. 213g xxx 11. 313hxx 12. xyx 13. 3432f xxxx 14. 3223xyx 15. 22321xxfxx 16. 23211g xxxx 17. 22325yxxx 18. 2521xfxx 19. 94yx 20. 1xfxx 21. 94yx 22. 234332yxxx 23. 2232xxyx 24. 2213xxyx Worksheet # 12: Higher Derivatives and Trigonometric Functions1. Calculate the indicated derivative:(a)f(4)(1),f(x) =x4(b)g(3)(5),g(x) = 2x2 x+ 4(c)h(3)(t),h(t) = 4et t3(d)s(2)(w),s(w) = wew2. Calculate the first three derivatives off(x) =xexand use these to guess a general formula forf(n)(x),then-th derivative Letf(t) =t+ 2 cos(t).(a) Find all values oftwhere the tangent line tofat the point (t,f(t)) is horizontal.

9 (b) What are the largest and smallest values for the slope of a tangent line to the graph off?4. Differentiate each of the following functions:(a)f(t) = cos(t)(b)g(u) =1cos(u)(c)r( ) = 3sin( )(d)s(t) = tan(t) + csc(t)(e)h(x) = sin(x) csc(x)(f)f(x) =x2sin(x)(g)g(x) = sec(x) + cot(x)5. Calculate the first five derivatives off(x) = sin(x). Then determinef(8)andf(37)6. Calculate the first 5 derivatives off(x) = 1/x. Can you guess a formula for thenth derivative,f(n)?7. A particle s distance from the origin (in meters) along thex-axis is modeled byp(t) = 2 sin(t) cos(t),wheretis measured in seconds.(a) Determine the particle s speed (speed is defined as the absolute value of velocity) at seconds.

10 (b) Is the particle moving towards or away from the origin at seconds? Explain.(c) Now, find the velocity of the particle at timet=3 2. Is the particle moving toward the origin oraway from the origin?(d) Is the particle speeding up at 2seconds?8. Find an equation of the tangent line at the point specified:(a)y=x3+ cos(x),x= 0(b)y= csc(x) cot(x),x= 4(c)y=e sec( ), = 49. Comprehension check for derivatives of trigonometric functions:(a) True or False: Iff ( ) = sin( ), thenf( ) = cos( ).(b) True or False: If is one of the non-right angles in a right triangle and sin( ) =23, then thehypotenuse of the triangle must have length Excel Supplemental Problems #121.