Calculus Team
March Statewide Invitational 2017
1
2
Let A = the value of
x
2
 x dx .
7
0
𝟐
𝟑
∫𝟎 𝒙𝒆−𝒙 𝒅𝒙.
7
1
1
Let C = the value of
Question #1
Let B = the value of
x
3
dx .
2
Let D = the value of :
7
 [3 f ( x)  2g( x)  1]dx, if  f ( x)dx  4 and  g( x)dx  2.
Calculus Team
1
March Statewide Invitational 2017
1
2
Let A = the value of
Let C = the value of
7
x
Question #1
2
 x dx .
0
𝟐
𝟑
∫𝟎 𝒙𝒆−𝒙 𝒅𝒙
Let B = the value of
x
3
2
.
Let D = the value of :
7
7
1
1
 [3 f ( x)  2g( x)  1]dx, if  f ( x)dx  4 and  g( x)dx  2.
1
dx .
Calculus Team
March Statewide Invitational 2017
Question #2
The function f is twice-differentiable and satisfies the conditions in the table below:
x
0
3
Let
f(x)
6
2
f /(x)
f // ( x )
2
4
1
6
g( x )  3sin(2 x )  f ( x ) and let h( x )  e f ( x ) .
A = the value of
g(0)
B = the value of
g/ (0)
h/ (3)
//
D = the value of h (3)
C = the value of
Calculus Team
March Statewide Invitational 2017
Question #2
The function f is twice-differentiable and satisfies the conditions in the table below:
x
0
3
Let
f(x)
6
2
f /(x)
f // ( x )
2
4
1
6
g( x )  3sin(2 x )  f ( x ) and let h( x )  e f ( x ) .
A = the value of
g(0)
B = the value of
g/ (0)
h/ (3)
//
D = the value of h (3)
C = the value of
Calculus Team
A = the value of
d
B( x) 
dx
C ( x) 
d
dx
x2

March Statewide Invitational 2017
Question #3
dx
for x 2 y  xy 2  x  3 when x  1 and y  0
dy
t dt , x  0
x
sin x
 2t
2
dt
x
2
d
2 x dy
D=
dx 0
Calculus Team
A = the value of
d
B( x) 
dx
C ( x) 
d
dx
2
D=
x2

March Statewide Invitational 2017
dx
2
2
for x y  xy  x  3 when x  1 and y  0
dy
t dt , x  0
x
sin x
 2t
x
d
2 x dy
dx 0
2
dt
Question #3
Calculus Team
March Statewide Invitational 2017
Question #4
Which of the following statements concerning the graph of f(x) = a𝑥 3 + b𝑥 2 + cx +d are true?
Assume a,b,c,d are non-zero rational numbers.
(Please write the entire word true or false for each part.)
A) The leftmost of the two extrema of the function is
−𝑏
3𝑎
−
√4𝑏2 −12𝑎𝑐
3𝑎
.
−𝑏
B) F(x) has a point of inflection at x = 3𝑎 .
C) The graph of f(x) is tangent to the x-axis at x = -1 if and only if 4a – 3b +2c – d = 0.
D) The distance along the x-axis between the 2 extrema of f(x) is
Calculus Team
√𝑏2 −3𝑎𝑐
3𝑎
.
March Statewide Invitational 2017
Question #4
Which of the following statements concerning the graph of f(x) = a𝑥 3 + b𝑥 2 + cx +d are true?
Assume a,b,c,d are non-zero rational numbers.
(Please write the entire word true or false for each part.)
−𝑏
A) The leftmost of the two extrema of the function is 3𝑎 −
√4𝑏 2 −12𝑎𝑐
3𝑎
.
−𝑏
B) F(x) has a point of inflection at x = 3𝑎 .
C) The graph of f(x) is tangent to the x-axis at x = -1 if and only if 4a – 3b +2c - d = 0.
D) The distance along the x-axis between the 2 extrema of f(x) is
√𝑏2 −3𝑎𝑐
3𝑎
.
Calculus Team
March Statewide Invitational 2017
Question #5
A curve is given parametrically by the equations x = 3 - 4sin(t) and y = 4 + 3cos(t) for t in the
interval [0,2𝜋].
Let A = Identify the curve
Let B = all (x,y) coordinates at which the curve has a vertical tangent.
Let C = all (x,y) coordinates at which the curve has a horizontal tangent.
Let D = all t values at which the curve has a slope of 0.75.
Calculus Team
March Statewide Invitational 2017
Question #5
A curve is given parametrically by the equations x = 3 - 4sin(t) and y = 4 + 3cos(t) for t in the
interval [0,2𝜋].
Let A = Identify the curve
Let B = all (x,y) coordinates at which the curve has a vertical tangent.
Let C = all (x,y) coordinates at which the curve has a horizontal tangent.
Let D = all t values at which the curve has a slope of 0.75.
Calculus Team
March Statewide Invitational 2017
Question #6
|𝑥|−𝑥
A) lim
𝑥→1 𝑥−1
B) lim
𝑥→0
cos(𝑥)−1
𝑥
1
𝑥
1
3
( )−( )
C) lim
𝑥→3
𝑥−3
D) lim [1 + sin(4𝑥)]cot(𝑥)
𝑥→0
Calculus Team
March Statewide Invitational 2017
|𝑥|−𝑥
A) lim
𝑥→1 𝑥−1
B) lim
𝑥→0
cos(𝑥)−1
𝑥
C) lim
𝑥→3
1
𝑥
1
3
( )−( )
𝑥−3
D) lim [1 + sin(4𝑥)]cot(𝑥)
𝑥→0
Question #6
Calculus Team
March Statewide Invitational 2017
Question #7
A) A balloon rises straight up at 10 ft/sec. An observer is 40 ft west the spot where the
balloon left the ground. Find the rate of change (in radians/sec) of the balloon’s angle of
elevation when the balloon is 30 ft off the ground.
B) A 2nd observer of the same balloon is 10 feet closer to the spot where the balloon left the
ground. Find the rate of change (in radians/sec) of the balloon’s angle of elevation when
the balloon is 30 ft off the ground.
C) Yet a 3rd observer is standing a mere 10 ft from the spot where the balloon left the
ground. Find the rate of change (in radians/sec) of the balloon’s angle of elevation when
the balloon is 30 ft. off the ground.
D) A 4th observer is 120 ft east of the spot where the balloon left the ground. . Find the rate
of change (in radians/sec) of the balloon’s angle of elevation when the balloon is 50 ft off
the ground.
Calculus Team
March Statewide Invitational 2017
Question #7
A) A balloon rises straight up at 10 ft/sec. An observer is 40 ft west the spot where the
balloon left the ground. Find the rate of change (in radians/sec) of the balloon’s angle of
elevation when the balloon is 30 ft off the ground.
B) A 2nd observer of the same balloon is 10 feet closer to the spot where the balloon left the
ground. Find the rate of change (in radians/sec) of the balloon’s angle of elevation when
the balloon is 30 ft off the ground.
C) Yet a 3rd observer is standing a mere 10 ft from the spot where the balloon left the
ground. Find the rate of change (in radians/sec) of the balloon’s angle of elevation when
the balloon is 30 ft. off the ground.
D) A 4th observer is 120 ft east of the spot where the balloon left the ground. . Find the rate
of change (in radians/sec) of the balloon’s angle of elevation when the balloon is 50 ft off
the ground.
Calculus Team
March Statewide Invitational 2017
Question #8
Compute the area of each region enclosed by the graphs of the given equations.
A)
y = 𝑒 −𝑥
y = 𝑒𝑥
B)
y = 2x
y=2 - 4
C)
y = sin(x)
y = sin(2x)
D)
y = ln(x)
y=1–x
Calculus Team
𝑥
x = ln(3)
y=0
y=2
on interval [0,𝜋]
y=2
March Statewide Invitational 2017
Compute the area of each region enclosed by the graphs of the given equations.
A)
y = 𝑒 −𝑥
y = 𝑒𝑥
B)
y = 2x
y=2 - 4
y=0
C)
y = sin(x)
y = sin(2x)
on interval [0,𝜋]
D)
y = ln(x)
y=1–x
𝑥
x = ln(3)
y=2
y=2
Question #8
Calculus Team
March Statewide Invitational 2017
Question #9
Evaluate the following integrals:
−1 2
𝑥2
A) ∫−2
dx
0 1+cos(2𝑥)
B) ∫𝜋
2
2
dx
2
C) ∫1 𝑥 3 [ln(𝑥)]𝑑𝑥
3𝜋
𝜃
𝜋
D) ∫𝜋2 𝑐𝑜𝑡 5 (6 )𝑠𝑒𝑐 2 ( 6 ) 𝑑𝜃
Calculus Team
March Statewide Invitational 2017
Evaluate the following integrals:
−1 2
𝑥2
A) ∫−2
dx
0 1+cos(2𝑥)
B) ∫𝜋
2
2
dx
2
C) ∫1 𝑥 3 [ln(𝑥)]𝑑𝑥
3𝜋
𝜃
𝜋
D) ∫𝜋2 𝑐𝑜𝑡 5 (6 )𝑠𝑒𝑐 2 ( 6 ) 𝑑𝜃
Question #9
Calculus Team
March Statewide Invitational 2017
Question #10
Evaluate each of the following limits.
A) lim+
𝑥→0
cot(𝑥)
ln(𝑥)
B) lim+ tan(𝑥)ln(𝑥)
𝑥→0
C) lim
1
𝑥→0 𝑥 2
–
cos(3𝑥)
𝑥2
1+𝑥
D) lim ( 𝑥+2 )𝑥
𝑥→∞
Calculus Team
March Statewide Invitational 2017
Evaluate each of the following limits.
A) lim+
𝑥→0
cot(𝑥)
ln(𝑥)
B) lim+ tan(𝑥)ln(𝑥)
𝑥→0
1
C) lim 𝑥 2 –
𝑥→0
cos(3𝑥)
𝑥2
1+𝑥
D) lim ( 𝑥+2 )𝑥
𝑥→∞
Question #10
Calculus Team
March Statewide Invitational 2017
Question #11
A = the volume of the solid whose base is the circle x 2  y 2  9 and whose cross-sections
perpendicular to the x-axis are squares.
0
Let B = the value of
e x dx .
 x
 e  1
Let C = the volume of the solid of revolution whose base is bounded by the lines
f ( x)  1  x, g ( x)  x  1, and x  0 and whose cross-sections are semicircles
perpendicular to the x-axis.

Let D = the value of
e x dx .
 x
 e  1
Calculus Team
March Statewide Invitational 2017
Question #11
A = the volume of the solid whose base is the circle x 2  y 2  9 and whose cross-sections
perpendicular to the x-axis are squares.
0
Let B = the value of
e x dx .
 x
 e  1
Let C = the volume of the solid of revolution whose base is bounded by the lines
f ( x)  1  x, g ( x)  x  1, and x  0 and whose cross-sections are semicircles
perpendicular to the x-axis.

Let D = the value of
Calculus Team
e x dx .
 x
 e  1
March Statewide Invitational 2017
Question #12
A cylindrical tank is initially filled with water to a depth of 16 ft. A valve in the bottom is opened
and water flows out. The depth, h, of the water in the tank decreases at a rate proportional to the
square root of the depth; that is
dh
 k h , where k is a constant and 0  k  1.
dt
(Use the value of k found in A to calculate parts B and D.)
A = the value of
k if, after the valve is opened, the water falls to a depth of 12.25 ft. in 8 hours.
B = the number of hours after the valve was first opened to make the tank completely empty.
C = A second cylindrical tank with radius 5 cm is being filled with water at a rate of 3 cubic cm
per minute. How fast is the height increasing?
D = AB/C
Calculus Team
March Statewide Invitational 2017
Question #12
A cylindrical tank is initially filled with water to a depth of 16 ft. A valve in the bottom is opened
and water flows out. The depth, h, of the water in the tank decreases at a rate proportional to the
square root of the depth; that is
dh
 k h , where k is a constant and 0  k  1.
dt
(Use the value of k found in A to calculate parts B and D.)
A = the value of
k if, after the valve is opened, the water falls to a depth of 12.25 ft. in 8 hours.
B = the number of hours after the valve was first opened to make the tank completely empty.
C = A second cylindrical tank with radius 5 cm is being filled with water at a rate of 3 cubic cm
per minute. How fast is the height increasing?
D = AB/C
Calculus Team
March Statewide Invitational 2017
Question #13
A particle travels along the curve x(t)  2t  2,t  0 .
A) Determine the velocity of the particle at t  1 .
B) Determine the acceleration of the particle at t  1 .
C) Determine the speed of the particle at t = 1
D) Determine the rate of change of the speed at time t  1 .
Calculus Team
March Statewide Invitational 2017
A particle travels along the curve x(t)  2t  2,t  0 .
A) Determine the velocity of the particle at t  1 .
B) Determine the acceleration of the particle at t  1 .
C) Determine the speed of the particle at t = 1
D) Determine the rate of change of the speed at time t  1 .
Question #13
Calculus Team
If f ( x) 
A)
4x
8x2  1
[𝑓(√3)]
March Statewide Invitational 2017
Question #14
, then find the value of the following:
2
B) 25𝑓′(−√3 )
C)
lim 𝑓(𝑥)
𝑥→∞
√3
D) ∫0 𝑓(𝑥)√8𝑥 2 + 1 𝑑𝑥
Calculus Team
If f ( x) 
A)
4x
8x2  1
[𝑓(√3)]
March Statewide Invitational 2017
, then find the value of the following:
2
B) 25𝑓′(−√3 )
C)
lim 𝑓(𝑥)
𝑥→∞
√3
D) ∫0 𝑓(𝑥)√8𝑥 2 + 1 𝑑𝑥
Question #14