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Practice problems for Midterm #
- Let
!
f ( x ) =^ $ %^ &^ &^ x (^2) ( x ,"^ " x x 2 ,^ )< (^20) ,^0 # x x >< 2 2 a) Evaluate each limit (if it exists). Write DNE for the final answer on any part^ '^ &^ & for which you think the limit does not exist. Also fill in any intermediate blanks with the piece of the function you are using. (i) (ii) x limlim! 0 + f (( x ))= (^) x limlim! 0 + __________________________________ =______________ (iii) x "^0!^ f^ x =^ x "^0!^ = !
(iv)^ lim^ x "^0^ f^ ( x )^ =^ _______ !
(v)^ x^ lim"^2 +^ f^ ( x )^ =^ lim^ x "^2 +^ _________________^ =^ _______ !
(vi)^ lim^ x "^2 #^ f^ ( x )^ =^^ x lim"^2 #^ _________________^ =^ _______ !
lim x " 2 f ( x ) = _______ b) Using your answers from part a, determine whether or not f is continuous at x = 0 and at !
- The displacement (in meters) of an object moving in a straig^^ x^ =^2. Be sure to explain why or why not. ht line is given by !
s = 2 + 4 t + t^3 where ! (a) Find the average velocity over the time interval [1,3].^^ t is measured in seconds. (b) Find the instantaneous velocity at (c) Find the acceleration at t = 1. t = 1.
- Find the derivative of f ( x )= x^12 by using the definition of the derivative.
- Find the equation of the tangent line to the curve in #3 at x=2. 5. Find the derivative of
- Find the equation of the tangent line to the curve in #5 at x=4.^^ f^ ( x )^ =^ x by using the^ definition^ of the derivative.
- Is !
continuous everywhere by redefining it at a single point? If so, what p^^ f^ ( x )^ =^ x^2^ x +^^ " x^^^2 "^6 continuous everywhere? Why or why not? Can we make itoint and how would we define it? 8. Compute the following limits. Show all work. a) !
x^ lim " (^164) x^ # #^16 x b) !
lim x " 1 $ % & (^) x^1 # 1 + (^) x (^2) # 13 x + 2 ' ( ) c) !
lim t " 2^ t t^23 ##^48 d) !
x^ lim "#$ 32 xx #^2 + 5 1 e) !
lim x "# 32 xx $^2 + 5 1 f) !
lim x "# ( x^2 + 1 $ x )
g) !
x^ lim " 0 +^ x^ sin^ # $^ %^1 x & '^ ( h) !
lim v " (^4) | 44 ##^ vv | i) !
lim x "#^2 x 53 x + 3 x +^2 4 + x^3 + x 1^ $^1 i) !
lim x "# 5 x (^3) + 36 xx^22 + (^) $^1 7 x + 1 j) !
lim x "#^ x 52 $+ xx k) !
lim h " 0 (#^3 +^ hh )^2 #^9 l) lim x " (^515) x^ ##^15^ x
- exist, explain why. The function f is graphed below. Find the following limits. If a limit does not
a) !
b)^ x^ lim^ "^0 +^ f^ ( x ) !
c)^ x^ lim^ "^0 #^ f^ ( x ) !
d)^ lim^ x^ "^0 f^ ( x ) !
e)^ x^ lim^ "^2 +^ f^ ( x ) !
f)^ x^ lim^ "^2 #^ f^ ( x ) !
g)^ lim^ x^ "^2 f^ ( x ) !
h)^ x^ lim^ "#^2 +^ f^ ( x ) !
i)^ x^ lim^ "#^2 #^ f^ ( x ) !
x^ lim "# 2 f^ ( x )
- Find all points of discontinuity for the function f in the interval ( graphed above in #13. Be sure to explain your answers. -3,3)
- Differentiate. (a) !
y = 5 ( 2 " x^3 ) 4 x 26 + 1
(b) y = (^) ( x^2 + (^1) ) (^) ( x + (^1) )^3
(c) !
y = x^92 x (^) +^ "^ x^5 x (d) !
y = ( x^2 " 7 ) # $ % x x^3 + 2 +^8 2 x^2 & ' (
(e) !
y = x^^3 x^42 ++^ x 35
- Find !
f "( x )in two different ways, where !
- Find the n-th derivative,^^ f^ ( x )^ =^ (^ x^ +^6 x )^ #^ $^ %^2 x^2 "^6^ x &^ '^ (. !
f ( n^ )( x ), of the function !
- Let^^ f^ ( x )^ =^ x^ +^3. !
h ( x ) = f ( g ( x )), !
g ( 4 ) = 2 , g ( 1 ) = 3 , f "( 3 ) = # 3 , g " ( 1 ) = 2 , g " ( 4 ) = 3 , !
f "(2) = #. Find
!
h "( 1 ) and !
- Find the equation of the tangent line to the curve a)^ h^ "^ (^4 ). !
y = x^3 g ( x )at x=1 and b) y = (^) gf^ (( xx )) if !
g ( 1 ) = 2 , f ( 1 ) = 3 , f "( 1 ) = 1 , g " ( 1 ) = # 2.
