MAT016A MT1 W10 Temple(2)

# MAT016A MT1 W10 Temple(2) - f-1(c(4 pts Evaluate f-1(1 5...

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MIDTERM EXAM I Math 16A Temple-Winter 2010 - Print your name, section number and put your signa- ture on the upper right-hand corner of this exam. Write only on the exam. - Show all of your work, and justify your answers for full credit. SCORES #1 #2 #3 #4 #5 #6 #7 #8 TOTAL:

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1. Determine the following limits: (a) (6 pts) lim x 2 r x 2 - 4 x - 2 (Hint: Factor) (b) (6 pts) lim x 0 x +2 - 2 4 x (Hint: Conjugate)
2. Determine the following limits: (c) (6 pts) lim x + 2 x 5 +3 x - 4 3 x 5 - 7 (Hint: Divide by highest power) (d) (6 pts) lim x 0 1 - cos 2 x tan 2 x cos x (Hint: Simplify)

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3. Let f ( x ) = 4 - x 2 , g ( x ) = 2 sin x . (a) (5 pts) Find the Domain of f . (b) (5 pts) Find the (precise) Range of g . (c) (6 pts) Find ( f g )( x ) . (d) (5 pts) Prove that ( f g )( x ) is defined for every real number x. (I.e, show the Domain of f g is all of R . )
4. Consider the function f ( x ) = 2 x +1 x - 3 with Domain x 6 = 3. (a) (7 pts) Find a formula for f - 1 ( x ). (b) (4 pts) Find the Domain of

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Unformatted text preview: f-1 . (c) (4 pts) Evaluate f-1 (1) 5. Find all vertical and horizontal asymptotes (you needn’t graph the functions): (a) (3 pts) y = sin x (b) (6 pts) y = 2 x 2 ( x 2-1)( x +2) (c) (6 pts) y = tan x 6. Given a function f ( x ): (a) (4 pts) State the deﬁnition of the derivative f ( x ) . (b) (7 pts) Directly from the deﬁnition, derive the value of f (2) if f ( x ) = x 3 . (Hint: ( x-a ) 3 = x 3-3 ax 2 +3 a 2 x-a 3 ). 7. (7 pts) Draw the graph of a function continuous except at x = 2 , such that f (2) = 1 , lim x → 2-= 0 , and lim x → 2+ = 4 . (Use closed and open dots correctly at x = 2 . ) 8. (7 pts) Find the equation of the line perpendicular to the graph of y = 2 x 2-3 at the point (1 ,-1)....
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