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MATH 1610 PRACTICE MIDTERM EXAM
DNE means “Does Not Exist”
Calculators are Not Allowed
- Evaluate lim x→ 0 x csc(3x). a) 0 b) DNE c)
d) 3 e) ∞ f) −∞ g) 1
- The displacement of a particle moving in a straight line is given by s = t^2 + t. Find the average velocity from t = 1 to t = 3. a) 5 b) 6 c) 7 d) 8 e) 9 f) 10
- Find the second derivative of y = sin(2x). a) 2 cos(2x) b) − sin(2x)
c) −2 sin(2x) d) −4 sin(2x) e) 4 sin(2x) f) 4 cos(2x)
- Evaluate (^) x→−lim 4 −
x x + 4
. a) −
b) DNE c) +∞ d) −∞ e)
- Evaluate lim x→ 1
x^3 − x x − 1
. a) DNE b)
c) 1 d) 2 e) 3 f) 4
- Find the derivative of f (x) = 12x^1 /^3 at x = 8.
a) 0 b) 1 c) 2 d) 3 e) 4
- Find the slope of the line that is normal to y =
( x^2 + 6x
) 1 / 2 at x = 2. a) − 2 b) − 1 c) 0 d) 1 e) 2
- Evaluate lim x→∞ 1 + x − 2 x^3 4 x^3 + x^2 + 1
a)
b) DNE c)
d) −
e) 0
- Find the derivative of y = csc (x^2 ).
a) − csc (x^2 ) cot (x^2 ) b) − 2 x csc (x^2 ) cot (x^2 ) c) 2x csc (x^2 ) cot (x^2 )
d) 2x csc^3 (x^2 ) e) − 2 x csc^3 (x^2 )
- Find the derivative of y = csc^2 (x).
a) 2 csc(x) b) 2 csc^3 (x) c) −2 csc^3 (x)
d) −2 csc^2 (x) cot(x) e) 2 csc^2 (x) cot(x)
- Find the derivative (dy/dx or y′) for the curve 2x^3 + 2y^3 − 9 xy = 0 at (1, 2).
a) 3 b) 9 c)
d)
e)
f)
g) 7 h) −
- Find the derivative of y = e^2 x 1 + x a) e^2 x^ b) (1 + x)2xe^2 x−^1 − e^2 x (1 + x)^2
c) 2e^2 x^ d) e^2 x(1 + 2x) (1 + x)^2
e) xe^2 x (1 + x)^2
- Find the derivative of y = xArctan(x)
a)
1 + x^2 b) tan−^1 (x) − x tan−^2 (x) sec^2 (x) c) − tan−^2 (x) sec^2 (x)
d) Arctan(x) + √ x 1 − x^2
e) Arctan(x) + x 1 + x^2
- Find lim h→ 0
1 2+h −^
1 2 h
. a) DNE b)
c) 1 d) − 1 e)
f) −
- Find the equation of the line that is tangent to the curve y = x − x^2 2 − x
at x = 4.
a) y = x 2
c) y = − 2 x +
d) y = − 2 x + 14 e) y = 2x + 2 f) y = 2x −
- Evaluate lim x→ 4 √^ x^ −^4 x − 2 a) 0 b) 1 c) 2 d) 3 e) 4 f) 5 g) DNE
Answers
- c 2. a 3. d 4. c 5. d 6. b 7. b 8. d
- b 10. d 11. e 12. d 13. e 14. f 15. a 16. e
- Find the derivative of y = cos (x^3 ).
a) − 3 x^2 sin (x^3 ) b) 3x^2 sin (x^3 ) c) 3 sin^2 (x)
d) 3 cos^2 (x) sin (x) e) −3 cos^2 (x) sin (x)
- Find the derivative (dy/dx or y′) for the curve x^2 y^3 − y^2 + xy − 1 = 0 at (1, 1).
a) 3 b) 2 c)
d)
e) −
f)
g) 1 h) −
- Find the derivative of y = 3 x^2 − x x + 1 at x = 1. a) − 2 b) − 1 c) 0 d) 1 e) 2 f) 3 g) DNE
- Find the derivative of y = exArcsec(x)
a) − sec−^2 x sec x tan x b)
ex |x|
x^2 − 1
+exArcsec(x) c) ex^ sec x tan x+Arcsec(x)ex
d) −ex |x|
x^2 − 1
x^2 − 1
- Find lim h→ 0
1 3+h −^
1 3 h
. a) DNE b)
c)
d) −
e)
f) −
- Find the equation of the line that is tangent to the curve y =
x at x = 4. a) y = x 4
- 1 b) y = x − 2 c) y = x 4
d) y = x 4 − 1 e) y = x 4
- Evaluate lim x→ 4
x − 2) 4 − x a) − 2 b) − 1 c) 0 d) 1 e) 2 f) 3 g) DNE
Answers
- a 2. b 3. b 4. c 5. f 6. b 7. d 8. e
- e 10. a 11. h 12. e 13. b 14. f 15. a 16. b
MATH 1610 PRACTICE MIDTERM EXAM 3
DNE means “Does Not Exist”
Calculators are Not Allowed
- Evaluate lim x→ 0 3 x cot(5x). a) 0 b) DNE c)
d) 3 e) ∞ f) −∞ g)
- The displacement of a particle moving in a straight line is given by s = t^3 − 2 t + 1. Find the instantaneous velocity at t = 2. a) 5 b) 6 c) 7 d) 8 e) 9 f) 10
- Find the second derivative of y = ex^ cos(x). a) ex^ cos(x)−ex^ sin(x) b) − 2 ex^ sin(x)
c) 2ex^ sin(x) d) 2ex^ cos(x) e) − 2 ex^ cos(x) f) ex^ sin(x) − x cos(x)
- Evaluate lim x→ 2 +
1 − x^3 x^2 (x − 2)
. a) −
b) DNE c) +∞ d) −∞ e)
- Evaluate lim x→ 3 x^2 − 2 x − 3 9 − x^2
a) DNE b)
c) 0 d) −
e) −
f) − 4
- Find the derivative of f (x) = 16x^1 /^2 at x = 16.
a) 0 b) 1 c) 2 d) 3 e) 4
- Find the slope of the line that is tangent to y = −
( x^3 − 4
) 1 / 2 at x = 2. a) − 2 b) − 1 c) 0 d) 1 e) 2
- Evaluate (^) x→−∞lim 1 − 2 x + 3x^2 3 − x^2 + 2x
a)
b) DNE c) −∞ d) 3 e) − 3
- Find the derivative of y = cot^2 (x).
a) − 2 x csc^2 (x^2 ) b) −2 cot^2 (x) csc (x) c) 2 csc^2 (x)
d) −2 cot (x) csc^2 (x) e) − 2 x csc (x^2 ) cot (x^2 )
MATH 1610 PRACTICE MIDTERM EXAM 4
DNE means “Does Not Exist”
Calculators are Not Allowed
- Evaluate lim x→ 0 x^2 csc(x^2 ). a) 0 b) DNE c)
d) 2 e) ∞ f) −∞ g) 1
- The displacement of a particle moving in a straight line is given by s = t^2 − t. Find the average velocity from t = 1 to t = 3. a) 3 b) 4 c) 5 d) 6 e) 7 f) 8
- Find the second derivative of y = e^2 x. a) 2e^2 x^ b) 2xe^2 x−^1
c) 2xe^2 x−^1 + 2e^2 x−^1 d) 2e^4 x^ e) 4e^4 x^ f) 4e^2 x
- Evaluate lim x→ 2 −
x^2 − 1 (x − 2)^2
. a)
b) DNE c) +∞ d) −∞ e)
- Evaluate lim x→ 2 x^2 − 4 x^2 − x − 2
a) DNE b)
c) 0 d)
e)
f) 4
- Find the derivative of f (x) = 4x^1 /^2 at x = 4.
a) 0 b) 1 c) 2 d) 3 e) 4
- Find the slope of the line that is tangent to y =
( x^2 + 2
) 2 at x = 2. a) − 2 b) − 1 c) 0 d) 1 e) 2
- Evaluate (^) x→−∞lim x^2 + 2x − 1 3 x + 6x^4
a)
b) DNE c)
d) −∞ e) 0
- Find the derivative of y = sec (x^2 ).
a) sec (2x) tan (2x) b) 2 sec (x) tan^2 (x) c) 2x sec (x^2 ) tan (x^2 )
d) 2 sec^2 (x) tan (x) e) − sec (2x) tan (2x)
- Find the derivative of y = sec^2 (x).
a) sec (2x) tan (2x) b) 2 sec (x) tan^2 (x) c) 2x sec (x^2 ) tan (x^2 )
d) 2 sec^2 (x) tan (x) e) − sec (2x) tan (2x)
- Find the derivative (dy/dx or y′) for the curve (x − y − 1)^3 = x at (1, −1).
a) 3 b) 2 c)
d)
e) −
f)
g) 1 h) −
- Find the derivative of y = x^2 − 2 2 − x at x = 1. a) − 2 b) − 1 c) 0 d) 1 e) 2 f) 3 g) DNE
- Find the derivative of y = cot(x)Arctan(x)
a)
1 + x^2 − csc(x) cot(x) b) − csc x cot xArctanx + cot(x) 1 + x^2 c) sec(x) cot(x) 1 + x^2
d) − csc(Arctan(x)) cot(Arctan(x)) ·
1 + x^2 e) csc x cot xArctanx − cot(x) 1 + x^2
- f (x) = 3 3
x^2. Find lim h→ 0 f (8 + h) − f (8) h
a) DNE b)
c) 1 d)
e) 2 f) 3
- Find the equation of the line that is tangent to the curve y = x + 2 3 − x at x = 2. a) y = 3x − 2 b) y = 5x − 6 c) y = 3x − 6
d) y = 5x − 2 e) y = 4x + 3 f) y = 4x − 3
- Evaluate lim x→ 2 x^2 − 4 x − 3 a) 0 b) 1 c) 2 d) 3 e) 4 f) 5 g) DNE
Answers
- g 2. a 3. f 4. d 5. d 6. b 7. d 8. e
- c 10. d 11. f 12. f 13. b 14. c 15. b 16. a
- Find the derivative of y = 13 tan^3 (x).
a) x^2 sec^2 (x^3 ) b) x^2 sec (x^3 ) tan (x^3 ) c) 13 sec (3x^2 )
d) tan^2 (x) sec^2 (x) e) 13 cot^2 (3x^2 )
- Find the derivative (dy/dx or y′) for the curve 5x^2 + 4y^2 − 16 xy = 16 at (0, 2).
a) 3 b) 2 c)
d)
e) −
f)
g) 1 h) −
- Find the derivative of y = e^3 x x − 1
a)
e^3 x(3x − 4) (1 + x)^2 b)
(x − 1)3xe^3 x−^1 − e^3 x (x − 1)^2 c)
e^3 x x − 1 d) 3e^3 x^ e)
xe^3 x (1 + x)^2
- Find the derivative of y = Arctan(ln(x))
a) sec^2 (x)ln(x) + tan(x) x
b)
(1 + ln^2 (x))x
c) Arctan(x) x
ln(x) 1 + x^2
d) Arctan(x) x
ln(x) 1 + x^2 e) − tan−^2 (x)ln(x) + Arctan(x) x
- Find lim h→ 0
ln(^14 + h) − ln(^14 ) h
a) DNE b)
c)
d) 1 e) 4 f) ln(^14 )
- Find the equation of the line that is tangent to the curve y = (x + 2)^6 at x = − 3.
a) y = 6x + 19 b) y = − 6 x − 19 c) y = 6x − 17
d) y = − 6 x − 17 e) y = − 6 x + 19 f) y = − 6 x + 17
- Evaluate lim x→ 1
x − 1) 1 − x a) − 3 b) − 2 c) − 1 d) 0 e) 1 f) 2 g) DNE
Answers
- a 2. b 3. c 4. c 5. f 6. d 7. b 8. d
- a 10. d 11. b 12. a 13. b 14. e 15. d 16. c
MATH 1610 PRACTICE MIDTERM EXAM 6
DNE means “Does Not Exist”
Calculators are Not Allowed
- Evaluate lim x→ 0 x^2 cot^2 (x). a) 0 b) DNE c)
d) 2 e) ∞ f) −∞ g) 1
- The displacement of a particle moving in a straight line is given by s = 2t^2 − t. Find the average velocity from t = 1 to t = 2. a) 3 b) 4 c) 5 d) 6 e) 7 f) 8
- Find the second derivative of y = ln(3x). a) arcsin(x)(
3 x
) b)
x
c) −
x^2 d) −
3 x^2 e)
x f) 3 ln(x) +
x
- Evaluate lim x→− 1 −
x x^2 + 3x + 2
. a) −
b) DNE c) +∞ d) −∞ e)
- Evaluate (^) xlim→− 1 x^2 + 3x + 2 x^2 − x − 2
a) DNE b)
c) 0 d) 3 e) −
f) − 3
- Find the derivative of f (x) = 83 x^3 /^4 at x = 16.
a) 0 b) 1 c) 2 d) 3 e) 4
- Find the slope of the line that is tangent to y =
( 17 x^2 − x
) 1 / 4 at x = 1. a) − 2 b) − 1 c) 0 d) 1 e) 2
- Evaluate lim x→∞ 9 x^3 + 5x^2 − 2 x − 3 x^2 − 5 x^3
a)
b) DNE c) −
d) −
e) 9
- Find the derivative of y = sin^2 (x).
a) sin (2x) b) 2x cos (x^2 ) c) 2 sin (x) cos (x)
d) −2 sin (x) cos (x) e) − 2 x cos (x^2 )
MATH 1610 PRACTICE MIDTERM EXAM 7
DNE means “Does Not Exist”
Calculators are Not Allowed
- Evaluate lim x→ 0 tan(2x) x . a) 0 b) DNE c)
d) 2 e) ∞ f) −∞ g) 1
- The displacement of a particle moving in a straight line is given by s = t^3 − 3 t + 1. Find the speed when the acceleration is zero. a) -3 b 0 c) 1 d) 2 e) 3
- Find the second derivative of y = arctan x
a) (^) 1+^1 x 2 b) (^) 1+−xx 2 c) (^) (1+−^2 xx (^2) ) 2 d) (^) (1+xx (^2) ) 2 e) (^) 1+−^2 xx 2
- Evaluate lim x→ 2 −
x + 1 x^3 (x − 2) a) 30 b) DNE c) -∞ d) ∞ e) (^00)
- Evaluate (^) xlim→− 6 x^2 + 11x + 30 x + 6 a) DNE b) 0 c) -1 d) 2 e) 3
- Find the derivative of f (x) = −^34 x (^43) at x = 8. a) -2 b) -1 c) 0 d) 1 e) 2
- Find the slope of the line that is tangent to y = 3x ln(2x) at x = (^12)
a) 0 b) 1 c) 2 d) 3 e) 4
- Evaluate lim x→∞ 3 x^2 − x^4 2 x − x^3 + 2x^2 a) 32 b) -1 c) 1 d) 12 e) ∞
- Find the derivative of y = cot(x^4 )
a) 4x^3 csc^2 (x^4 ) b) − 4 x^2 csc^2 (x^4 ) c)4 csc^2 (x^4 ) d) − 4 x^3 csc(x^4 ) e) − 4 x^3 csc(x)
- Find the derivative of y = cot^4 (x)
a) 4 csc^2 (x) cot^3 (x) b) −4 csc(x) cot^3 (x) c)4 csc^2 (x) cot(x) d) −4 csc^2 (x) cot^3 (x) e)4 cot^3 (x)
- Find the derivative of drdθ for the curve rθ^3 + r^3 θ + θ^3 = 3 at (1, 0)
a) -3 b) 0 c)1 d) 2 e)
- Find the derivative of x^2 arccos x
a) arcsin x b) √ 1 x−^3 x 2 c)2x arccos x − √ 1 x−^3 x 2 d) 2x arccos x + √ 1 x−^3 x 2 e)x^2 arccos x + x √^3 1 −x^2
- Find the derivative of y = x^2 − 1 x^2 + 1 at x = 1 a) -1 b) 0 c)1 d) 2 e)
- f (x) = ex^2 −^1. Find lim h→ 0 f (1 + h) − f (1) h a) 0 b) 1 c)2 d) 3 e) 4
- Find the equation of the tangent line to the curve y = ex^ sin x at x = 0.
a) y = −x + 1 b) y = x c)y = x + 1 d) y = 2x − 1 e) y = x + 2
- Evaluate lim x→ 25
x) x − 25 a) -1 b) 0 c)1 d) 2 e) 4
Answers
- d 2. e 3. c 4. c 5. c 6. d 7. d 8. e
- d 10. d 11. d 12. a 13. c 14. c 15. b 16. a
- Find the derivative ( dydx or y′) for the curve 2x^3 + 2y^3 − 9 xy = 0 at (1, 1).
(a) 7 (b) −^12 (c) 12 (d) -1 (e) -7 (f) DNE
- Find the derivative of y = e 2 x 3+x. (a) e
2 x(5+2x) (3+x)^2 (b)^ e
2 x (^) (c) (3+x)2xe^2 x−^1 −e^2 x (3+x)^2 (d)^
xe^2 x (3+x)^2
- Find the derivative of y = xarccot(x).
(a) (^) 1+−^1 x 2 (b) cot−^1 (x) − x cot−^2 (x) (c) arccot(x) + √ 1 −−xx 2 (d) arccot(x) − (^) 1+xx 2
- Find lim h→ 0
1 3+h −^
1 3 h
(a) DNE (b) 00 (c) 1 (d) -1 (e) −^19 (f) (^19)
- Find the equation of the line that is tangent to the curve y = x 2 −−xx^2 at x = 3.
(a) y = −x + 9 (b) y = − 1 (c) y = x 2 − 32 (d) y = − 2 x + 3 (e) y = 2x + 2
- Evaluate lim x→ 9 x − 9 √ x − 3
(a) 0 (b) DNE (c) 6 (d) 5 (e) 4 (f) 3
Answers
- c 2. a 3. d 4. c 5. b 6. c 7. e 8. c
- e 10. b 11. f 12. a 13. d 14. e 15. a 16. c
MATH 1610 Practice Midterm 9
- Evaluate lim x→ 0 5 tan x 6 x
- The displacement of a particle moving in a straight line is given by s = t^3 − 3 t + 4. Find the average velocity from t = 2 to t = 4, the velocity when t = 3, and the acceleration when t = 4.
- Find the second derivative of y = x cos 2x.
- Evaluate (^) xlim→ 3 + x + 2 x(x − 3)
- Evaluate lim x→ 3 x^2 + 2x − 15 x^2 − 9
- Find the derivative of 8x (^34) at x = 16.
- Find the slope that is normal to the curve y = ln(3x^3 + 4x) at x = 2.
- Evaluate limx→∞ x^3 + 4x − 15 8 x + x^4 − 9
- Find the derivative of y = sec(x^3 ).
- Find the derivative of y = sec^3 x.
- Find the derivative ( dydx or y′) for the curve x^2 + 2xy − y^2 + x = 2 at (1, 2).
- Find the derivative of y = 1 −^2 x 2 4 x+.
- Find the derivative of y = arctan(ex).
- Let f (x) = (x + 3)^2. Find lim h→ 0
f (2 + h) − f (2) h
- Find the equation of the line that is tangent to the curve y = x x+1+2 at x = 3.
- Evaluate lim x→ 4
x + 5 − 3 4 − x
Answers:
- (^56)
- avg.vel = 25, v = 24, a = 24
- −4(x cos 2x + sin 2x)
- ∞
- (^43)
- 3
- − (^45)
- 0
- 3x^2 sec(x^3 ) tan(x^3 )
- 3 sec^3 (x) tan x
- (^72)
- −4(x (^2) +4x+1) (4x+4)^4
- (^) 1+exe 2 x
- 10
- y = 251 x + (^1725)
- − (^16)
