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Calculus 1 Exam#1
Your name:
Date
[(Q1.) to (Q10.) Multiple Choice] 4 pts each. Circle your answer. No partial credit.
x −3
(Q1.) Which of the following about f ( x ) = 2
is true?
x − 4x + 3
(A) f (3) = 0
(B) The domain of f is (−∞,1)∪(1,∞)
(C) f has one horizontal asymptote at y = 0
(D) f has two vertical asymptotes at x = 1 and x = 3
(E) none of the above
(Q2.)
d ⎛ x ⎞
=?
dx ⎝ x 2 + 1 ⎠
1
(A)
2x
−x 2 + 1
(B) 2
x +1
1
(C) 2
( x + 1)2
−x 2 + 1
(D) 2
( x + 1)2
1
=?
x →7 ln( x − 7 )
(A) ∞
(B) −∞
(C) 0
(D) 1
(E) does not exist
(Q3.) lim
+
1
(Q4.) Which of the following function is not continuous everywhere ?
2
if x ≤ 0
⎪⎧− x
(A) f ( x ) = ⎨ 2
if x > 0
⎩⎪ x
⎧2 x + 1 if x ≤ 0
(B) f ( x ) = ⎨
⎩−2 x + 1 if x > 0
if x ≤ 0
⎧1
(C) f ( x ) = ⎨ x
if x > 0
⎩e
if x ≤ 0
⎧sin x
(D) f ( x ) = ⎨
⎩cos x if x > 0
dy
1
=?
(Q5.) If y = 4 , then
x
dx
1
(A)
4x 3
−4
(B) 5
x
−1
(C)
4x 5
4
(D) 3
x
x 5 − 32
represents…
x →2 x − 2
(A) f ′(2) , where f ( x ) = x 5
(B) f ′(32) , where f ( x ) = x 5
x 5 − 32
′
f
(2)
f
(
x
)
=
(C)
, where
x −2
x 5 − 32
′
(D) f (32) , where f ( x ) =
x −2
(E) none of the above
(Q6.) lim
y = f (x )
(Q7.) Given the graph of y = f ( x )
Which of the following statement is true?
(A) f ( π2 ) = 0
(B) lim f ( x ) = 2
x → π2
(C) lim f ( x ) does not exist
x →∞
(D) the domain of f is (−∞, π2 )∪( π2 ,∞)
(E) none of the above
2
(Q8.)
d 3 x
(x e ) = ?
dx
(A) 3 x 2e x
(B) x 3 + e x
(C) 3 x 2 + e x
(D) x 3e x + 3 x 2e x
(E) none of the above
(Q9.) Which of the following about f ( x ) = x is false?
(A) the domain of f is (−∞,∞)
(B) f is continuous on (−∞,∞)
(C) f is differentiable on (−∞,∞)
(D) none of the above
−1
(Q10.) lim tan
x →∞
(e ) = ?
1
x
(A) 0
(B) π
4
(C) π
2
(D) 1
(E) ∞
[(Q11.) to (Q20.) Show Your Work] points as indicated, write your answers on the blanks
(Q11.) A particle is moving on a straight line and its position is given by s(t ) = −2t 3 + 17t , where t
is measured in seconds and s in meters. What is the instantaneous velocity of the particle at t
= 5?
3
(Q12.) What are the 3 types of discontinuities? Draw a picture to illustrate each type.
(Q13.) [3 pts each] Given the graph of y = f ( x ) on the right. Determine the followings
y = f (x )
f (1) =
f ′(1) =
⎧ x 2 + 2x − 8
if x ≠ 2
⎪
(Q14.) [3 pts each] Determine the followings, where f ( x ) = ⎨ x − 2
⎪⎩5
if x = 2
(a) f (2) = adfddfdff lim d
δ x →0
(b) lim f ( x ) = adfddfdff lim d
x →2
δ x →0
4
(Q15.) Let f ( x ) = 3 x + 1 . Use a limit definition of derivative to obtain f ′(8) .
0 points if you use a different method.
(Q16.) Sketch a graph for y = f ( x ) so that it has the following properties
lim f ( x ) = ∞
x →2
lim f ( x ) = ∞
x →∞
lim f ( x ) = ∞
x →−∞
(Q17.) Evaluate and show all the algebraic steps.
x 3 −64
lim
=
x →4 x − 4
5
(Q18.) Evaluate and show all the algebraic steps.
1
−1
lim 3+h 3 =
h→0
h
(Q19.)
d ⎛⎜ x ⎞⎟
⎟=
⎜
dx ⎜⎝ x + 4 ⎟⎟⎠
sin(3 x )
x →0
x
(Q20.) Use a table of values to determine lim
End of Exam! Your Score is _______________ / 100
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