Calculus 1000A Summer 2023 Practice Midterm
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Calculus 1000A
Summer 2023
Practice Midterm
Multiple Choice Questions
1. The function f(x) = loga (x + 1) + b whose graph passes through points P(0, 2) and Q(8, 0) is?
Answer: f(x) = log1/3(x + 1) + 2.
2. The function f(x) = 4 − abx whose graph passes through the point P(2, 0) is?
Answer: f(x) = 4 − 2x .
3. Find the function f− 1 (x), if f(x) = 3(lnx)5 .
Answer: f − 1 (x) = e ^5x/3 .
4. Find the function f− 1 (x), if f(x) = cos(ln(^x)).
Answer: f − 1 (x) = e2 arccos x .
5. Find the domain of the function f(x) = sin− 1 (e2x).
Answer: ( −∞ , 0].
6. Find the domain of the function f(x) = ln(sin− 1 (x)).
Answer: (0, 1].
7. Evaluate tan(arcsin( 
)).
Answer: 4
8. Evaluate arcsin(cos( − 
)).
π
4 .
9. What inequalities are satisfied by the numbers x,y,z, if
x = log3 (log2 8), y = 3arccos(− 1), z = ?
Answer: x < z < y .
10. What inequalities are satisfied by the numbers x,y,z, if
x = π/2, y = eln 2023 , z = arctan(2023)?
Answer: z < x < y .
11. If lim = e3 , then lim f(x) = ?
x → 1 − x − 1 x →2 −
Answer: lim f(x) = −2.
ef(x)
12. If lim = 0, then lim f(x) = ?
Answer: lim f(x) = −∞ .
x →0
13. What are the horizontal and/or vertical asymptotes of the functions
(a) f(x) = (b) f(x) = (c) f(x) = ?
Answer:
(a) vertical {x = −2}; horizontal {y = 1}
(b) vertical {x = −2}; no horizontal
(c) vertical {x = −2} and {x = 1}; horizontal {y = 2}.
sin(2x)cos(3x)
Answer: 1
15. lim = ?
Answer: 6.
16. lim = ?
Answer: 0.
17. lim = ?
Answer: 2
Answer: 1
x → −∞ −3x2 + 2x − 1
Answer: −∞ .
1 − x
Answer: ∞ .
21. 
ln(sin − 1 ( 
) = ? Answer: 0.
22. 
arctan(e2023 ) = ? Answer: 0.
23. If 2x3 + 4y5 = 6, then 
= ?
3x2
10y4 .
dy
sec(xy) − y
25. 
3sin x = ?
Answer: 3sin x · ln3 · cos x.
26. 
cos(ln(^x)) = ?
Answer: − 2x .
27. 
ln(sin(ex )) = ? Answer: ex cot(ex ).
d
Answer: −2023tan x.
29. If f(x) = cos(x + 
) and g(x) is a function such that g\\ (0) = 2023 and (fg)\\ (0) = 4, then g\ (0) = ?
Answer: −2.
30. If f(x) = sin(2x), g(x) = cos(4x), and h(x) is a function such that h\ (π/4) = 3, then (fgh)\ (π/4) = ?
Answer: −3.
Short Answer Questions
1. Use the Squeeze Theorem to find lim (e− 1/北 · cos( 2北(02)3 )). Justify your answer.
2. Use the Squeeze Theorem to find lim (sin( π北 ) · ln(cos2 x + 1)). Justify your answer.
3. Use the Intermediate Value Theorem to show that the equation tan(x − 
) = 2cos 北
has a solution. Justify your answer.
4. Use the Intermediate Value Theorem to show that the equation
x3 + 2x + 1 = e −北 + 5
has a solution. Justify your answer.
5. Let A and B be real numbers and let
'
f(x) =〈
'
(
Aarcsin(esin 北 ) ,
,
2 − ^北+4
x < 0
x = 0
x > 0 .
(a) Find lim f(x) and lim f(x).
北 →0 − 北 →0+
(b) For what values of A and B is the function f continuous at 0? Justify your answer.
6. Let f(x) = logC (x2 ). Find the value of C for which the tangent line to the graph of f(x) at the
point P(1, 0) coincides with the tangent line to the curve (x − 2)2 + (y − 1)2 = 2 at the same
2023-06-21