Quiz 2 math 223
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mat223_quiz__2.pdf
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Quiz #2 (Group) MAT 223 Score: ( ) Last Name: ( ) First Name: ( ); Last Name: ( ) First Name: ( )
Last Name: ( ) First Name: ( ); Last Name: ( ) First Name: ( )
Show your all work. Good Luck!
1. (12 points) For 𝑓(𝑥) = ! ! solve each problem.
(1) Use the definition of derivative, find the 𝒇′(𝒙)
(2) Find the equation of tangent line to the curve at 𝑥 = 2
(3) Draw the graph of 𝑓(𝑥) and the equation of tangent line to the curve at 𝑥 = 2
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2. (12 points) For 𝑓(𝑥) = 𝑥 solve each problem.
(1) Use the definition of derivative, find the 𝒇′(𝒙)
(2) Find the equation of tangent line to the curve at 𝑥 = 4
(3) Draw the graph of 𝑓(𝑥) and the equation of tangent line to the curve at 𝑥 = 4
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3. (6 points) For 𝑓 𝑥 = 𝑥!, find the average rate of change.
(1) 𝑥 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑓𝑟𝑜𝑚 1 𝑡𝑜 3 (2) 𝑥 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑓𝑟𝑜𝑚 1 𝑡𝑜 2
4. (12 points) Use the Theorem of Limits to find the limits.
(1) lim!→! 𝑥! − 2𝑥! − 7 (2) lim!→! 𝑥! + 3𝑥! − 5𝑥 + 2
(3) lim!→!" !!! !!!"
(4) lim!→!! !!!!!!! !!!!!!!
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5. (14 points) For 𝑓 𝑥 = ! ! !!! , !"# !!!
!!!! , !"# !!!
(1) Draw the graph of 𝑓(𝑥)
(2) 𝑓 3 (3) lim!→!! 𝑓(𝑥) (4) lim!→!! 𝑓(𝑥)
(5) Use the definition of continuity, show that the function 𝒇 𝒙 is continuous 𝒂𝒕 𝒙 = 𝟑 or not?
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6. (8 points) Find the !" !"
(1) 𝑦 = 2𝑥!" + 4𝑥!! + 8 𝑥 (2) 𝑦 = ! !! − !
! + !!
!
7. (8 points) Solve each problem.
(1) Find of equation of tangent line to the graph of 𝑓 𝑥 = 𝑥! − 2𝑥 + 1 at 𝑥 = 1
(2) Find the points on the graph 𝑦 = ! ! 𝑥! − !
! 𝑥! + 3 at which the tangent line is horizontal.
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8. (16 points) Differentiate and simplify completely.
(1) 𝑦 = 4𝑥!(𝑥! + 5𝑥) (2) 𝑦 = !! !!!
!!!!
(3) 𝑦 = 𝑥! + 3𝑥 − 1 (4) 𝑦 = (5 − 2𝑥)!
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9. (12 points) Solve each problem.
(1) If 𝑦 = 7𝑢! + 2𝑢 and 𝑢 = 𝑥! + 1 , then find !" !"
(2) If 𝑓 𝑥 = 𝑥! and 𝑔 𝑥 = 3 𝑥 − 2 , then find 𝑓 ∘ 𝑔(−1)
(3) For 𝑦 = 𝑥! − 𝑥! + 5𝑥!, find ! !!
!!!
Good Luck to you.
