AP Calculus BC Study Guide Exam
Questions And Answers
Water leaking onto a floor forms a circular pool. The radius of the pool increases at a
rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?
Which of the following represents the notation of our final answer to this problem? -
Answer-dA/dt = 40pi cm²/min
/.Which of the following are acceptable guidelines for solving Related Rates problems? -
Answer--Label all quantities that can change as variables.
-Draw a picture.
-Find an equation that relates the variables.
^All of these are acceptable
/.A bicycle path and a road cross at right angles. A police woman stands on the road 70
meters south of the crossing and watches an eastbound cyclist traveling at 6 meters per
second. At how many meters per second is the cyclist moving away from the police
woman 4 seconds after passing the intersection? - Answer-1.95 m/sec.
/.The radius of a circle is increasing at a constant rate of 0.4 feet/sec. What is the rate of
increase in the area of the circle at the instant when the circumference of the circle is 40
pi feet - Answer-16pi ft²/second
/.A water tank in the shape of a right circular cone has a height of 10 feet. The top rim of
the tank is a circle with a radius of 4 feet. If water is being pumped into the tank at the
rate of 2 cubic feet per minute, what is the rate of change of the water depth, in feet per
minute, when the depth is 5 feet?
(The volume V of a right circular cone is V = 1/3 pi r2h, where r is its radius and h is its
height.) - Answer-1/(2pi) ft/min
/.The maximum value of
f(x) = x³ + 3x² - 9x - 2
on the interval [0, 2] is - Answer-0
/.Find the value of c guaranteed by the Mean Value Theorem for f(x) = 2/(x-1) on [3,5] -
Answer-x= 1 + (2√2)
/.Is Rolle's Theorem applicable to f(x) = 2/(x-1) on [3,5]? - Answer-No, f(a) does not
equal f(b)
/.Determine whether or not Rolle's Theorem applies. If it does, find all values of c in the
open interval (a,b) such that f'(c)=0
,f(x)= (2)/(x+1)²; [-3,0] - Answer-No the theorem does not apply
/.Which of the following statements is true: - Answer-The derivative of a function may be
used to determine whether or not the function is increasing or decreasing on any
intervals in its domain.
/.Find the differential dy of y= 2x - cot²(x) - Answer-(2 + 2 cot x csc² x)dx
/.Which of the following functions are differentiable for all values of x?
I. f(x)=|x|
II. f(x)=³√x
III. f(x)=x³ - Answer-III. only
/.Find f"(x) when f(x) = csc(x). - Answer-f"(x) = cscx(cot²x + csc² x)
/.Find f"(x) when f(x) = x * sin(x). - Answer-f"(x) = 2cos(x) - x * sin(x)
/.The position of a particle moving along the x-axis is given by x(t)=t³-5t²-8t+2 for t≥0.
The particle is at rest is at t= - Answer-4
/.If f(x) = (2x + 1)⁴, then the fourth derivative of f(x) at x = 0 is - Answer-384
/.Suppose that f(x) and g(x) are differentiable for all x with the values given in the table
below.
https://files.catbox.moe/z8gol2.png
Let h(x) = f(g(x)). Find h'(2). - Answer-20
/.If f(x) = (x+1)² cos(x), then f'(0) = - Answer-2
/.If dy/dx = 5xy, then (d²y)/(dx²) = - Answer-25x²y + 5y
/.If dy/dx = 2x + 2y, then (d²y)/(dx²) = - Answer-4x + 4y + 2
/.For the curve x + y = 3xy, which of the following is equal to dy/dx? - Answer-(3y-1)/(1-
3x)
/.If x³ + x * sin(y) = 1, then at point (1,0), dy/dx = - Answer--3
/.A function that is not differentiable at x = c must not be continuous at x = c. - Answer-
False
/.Find f"(x) when f(x) = x * cos(x). - Answer-f"(x) = -2sin(x) - x * cos(x)
/.If f(a) exists, them limx→a f(x) exists. - Answer-False
/.limx→0(((x+3)² - 9)/x) - Answer-6
,/.Which of the following points of discontinuity of https://files.catbox.moe/hce39h.png
is not removable? - Answer-x = 1
/.The function f is continuous on the closed interval [0,2] and has values given in the
table above. The equation f(x) = 1/2 must have at least two solutions in the interval [0,2]
if k =
https://files.catbox.moe/k04p63.png - Answer-0
/.Use the graph of f(x) to evaluate the limit.
https://files.catbox.moe/doizcg.png
lim x→2 lim f(x) = - Answer-undefined
/.Use the graph of f(x) to evaluate the limit.
https://files.catbox.moe/qiiosj.png
lim x→1⁺ f(x) = - Answer-2
/.Find the limit of:
https://files.catbox.moe/k7t3lc.png - Answer-0
/.The equation of the horizontal asymptote for the graph of y = (3-e¹/ˣ)/(3+e¹/ˣ) is -
Answer-y = 1/2
/.Solve for x:
5x-2>3 or x-7<1 - Answer-(−∞,∞)
/.|2x−8|<3 - Answer-(5/2, 11/2)
/.Find an equation for the line through (-3,4) and perpendicular to 5x+2y=8 - Answer-y =
(2/5)x + 26/5
/.Find an equation for the line through (2,7) and perpendicular to y = (4/5)x -3 - Answer-
5x + 4y = 38
/.What is the range of f(x) = x²-6x+5 - Answer-[−4,∞)
/.Given the graph of f(x) shown, which of the following represents f(2x)?
https://files.catbox.moe/z1dlto.PNG - Answer-https://files.catbox.moe/u9rc31.PNG
/.Given f(x) = x², which of the following represents f((1/2)x) - Answer-
https://files.catbox.moe/cmpft8.PNG
/.Use a calcuator to find the x-coordinates of the local extrema of f(x) = x³-8x²+5x+1 -
Answer-Local maximum at x = 1/3
Local minimum at x = 5
, /.lim x→2+ {2x-3, x<2 {x²+1, x≥2
piecewise: https://files.catbox.moe/lvseu5.PNG - Answer-5
/.Simplify the expression (sec x - tan x)(sin x + 1) - Answer-cos x
/.Simplify the expression sin(x) + cos(x)cot(x) - Answer-csc(x)
/.Given a data set, use a graphing calculator or other device to determine the line of
best fit (least-squares regression line).
Let x high temperature (in degrees F) and y the number of cups of lemonade sold at a
local stand.
https://files.catbox.moe/sinjrh.PNG
Use the best fit line to predict the number of cups of lemonade that will sell when the
high temperature is 98. - Answer-57
/.If lim x→a f(x) exists, then f(a) exists. - Answer-False
/.True or False:
If you know in advance that lim x→4 f(x) exists, you can determine the value of the limit
by knowing all the values of f(x) for all x > 4? - Answer-True
/.lim x→0 (1/(x+3) - 1/3)/x
Added image b/c it's hard to read
https://files.catbox.moe/kbl61f.PNG - Answer--1/9
/.lim x→2- {2x-3, x<2 {x²+1, x≥2
piecewise: https://files.catbox.moe/ua2frr.PNG - Answer-1
/.Use the graph of f(x) above to determine which one of the following statements is true.
https://files.catbox.moe/4s8uqk.PNG - Answer-lim x→1 f(x) exists.
/.Use the graph of f(x) to evaluate the limit. lim x→2 f(x) =
https://files.catbox.moe/3tqjma.PNG - Answer-1
/.lim x→0+ |x|/x =
This is the greatest integer function. - Answer-0
/.lim x→0- 1/x^2 - Answer-+∞
/.lim x→∞ tan⁻¹(x²-x⁴) = - Answer--π/2
/.Find any Horizontal Asymptotes of
g(x) = (x-3)/(x2-4) - Answer-y=0
/.Evaluate the limit:
lim x→∞ (5x³+10x²-2)/(8x⁴+1) - Answer-0
Questions And Answers
Water leaking onto a floor forms a circular pool. The radius of the pool increases at a
rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?
Which of the following represents the notation of our final answer to this problem? -
Answer-dA/dt = 40pi cm²/min
/.Which of the following are acceptable guidelines for solving Related Rates problems? -
Answer--Label all quantities that can change as variables.
-Draw a picture.
-Find an equation that relates the variables.
^All of these are acceptable
/.A bicycle path and a road cross at right angles. A police woman stands on the road 70
meters south of the crossing and watches an eastbound cyclist traveling at 6 meters per
second. At how many meters per second is the cyclist moving away from the police
woman 4 seconds after passing the intersection? - Answer-1.95 m/sec.
/.The radius of a circle is increasing at a constant rate of 0.4 feet/sec. What is the rate of
increase in the area of the circle at the instant when the circumference of the circle is 40
pi feet - Answer-16pi ft²/second
/.A water tank in the shape of a right circular cone has a height of 10 feet. The top rim of
the tank is a circle with a radius of 4 feet. If water is being pumped into the tank at the
rate of 2 cubic feet per minute, what is the rate of change of the water depth, in feet per
minute, when the depth is 5 feet?
(The volume V of a right circular cone is V = 1/3 pi r2h, where r is its radius and h is its
height.) - Answer-1/(2pi) ft/min
/.The maximum value of
f(x) = x³ + 3x² - 9x - 2
on the interval [0, 2] is - Answer-0
/.Find the value of c guaranteed by the Mean Value Theorem for f(x) = 2/(x-1) on [3,5] -
Answer-x= 1 + (2√2)
/.Is Rolle's Theorem applicable to f(x) = 2/(x-1) on [3,5]? - Answer-No, f(a) does not
equal f(b)
/.Determine whether or not Rolle's Theorem applies. If it does, find all values of c in the
open interval (a,b) such that f'(c)=0
,f(x)= (2)/(x+1)²; [-3,0] - Answer-No the theorem does not apply
/.Which of the following statements is true: - Answer-The derivative of a function may be
used to determine whether or not the function is increasing or decreasing on any
intervals in its domain.
/.Find the differential dy of y= 2x - cot²(x) - Answer-(2 + 2 cot x csc² x)dx
/.Which of the following functions are differentiable for all values of x?
I. f(x)=|x|
II. f(x)=³√x
III. f(x)=x³ - Answer-III. only
/.Find f"(x) when f(x) = csc(x). - Answer-f"(x) = cscx(cot²x + csc² x)
/.Find f"(x) when f(x) = x * sin(x). - Answer-f"(x) = 2cos(x) - x * sin(x)
/.The position of a particle moving along the x-axis is given by x(t)=t³-5t²-8t+2 for t≥0.
The particle is at rest is at t= - Answer-4
/.If f(x) = (2x + 1)⁴, then the fourth derivative of f(x) at x = 0 is - Answer-384
/.Suppose that f(x) and g(x) are differentiable for all x with the values given in the table
below.
https://files.catbox.moe/z8gol2.png
Let h(x) = f(g(x)). Find h'(2). - Answer-20
/.If f(x) = (x+1)² cos(x), then f'(0) = - Answer-2
/.If dy/dx = 5xy, then (d²y)/(dx²) = - Answer-25x²y + 5y
/.If dy/dx = 2x + 2y, then (d²y)/(dx²) = - Answer-4x + 4y + 2
/.For the curve x + y = 3xy, which of the following is equal to dy/dx? - Answer-(3y-1)/(1-
3x)
/.If x³ + x * sin(y) = 1, then at point (1,0), dy/dx = - Answer--3
/.A function that is not differentiable at x = c must not be continuous at x = c. - Answer-
False
/.Find f"(x) when f(x) = x * cos(x). - Answer-f"(x) = -2sin(x) - x * cos(x)
/.If f(a) exists, them limx→a f(x) exists. - Answer-False
/.limx→0(((x+3)² - 9)/x) - Answer-6
,/.Which of the following points of discontinuity of https://files.catbox.moe/hce39h.png
is not removable? - Answer-x = 1
/.The function f is continuous on the closed interval [0,2] and has values given in the
table above. The equation f(x) = 1/2 must have at least two solutions in the interval [0,2]
if k =
https://files.catbox.moe/k04p63.png - Answer-0
/.Use the graph of f(x) to evaluate the limit.
https://files.catbox.moe/doizcg.png
lim x→2 lim f(x) = - Answer-undefined
/.Use the graph of f(x) to evaluate the limit.
https://files.catbox.moe/qiiosj.png
lim x→1⁺ f(x) = - Answer-2
/.Find the limit of:
https://files.catbox.moe/k7t3lc.png - Answer-0
/.The equation of the horizontal asymptote for the graph of y = (3-e¹/ˣ)/(3+e¹/ˣ) is -
Answer-y = 1/2
/.Solve for x:
5x-2>3 or x-7<1 - Answer-(−∞,∞)
/.|2x−8|<3 - Answer-(5/2, 11/2)
/.Find an equation for the line through (-3,4) and perpendicular to 5x+2y=8 - Answer-y =
(2/5)x + 26/5
/.Find an equation for the line through (2,7) and perpendicular to y = (4/5)x -3 - Answer-
5x + 4y = 38
/.What is the range of f(x) = x²-6x+5 - Answer-[−4,∞)
/.Given the graph of f(x) shown, which of the following represents f(2x)?
https://files.catbox.moe/z1dlto.PNG - Answer-https://files.catbox.moe/u9rc31.PNG
/.Given f(x) = x², which of the following represents f((1/2)x) - Answer-
https://files.catbox.moe/cmpft8.PNG
/.Use a calcuator to find the x-coordinates of the local extrema of f(x) = x³-8x²+5x+1 -
Answer-Local maximum at x = 1/3
Local minimum at x = 5
, /.lim x→2+ {2x-3, x<2 {x²+1, x≥2
piecewise: https://files.catbox.moe/lvseu5.PNG - Answer-5
/.Simplify the expression (sec x - tan x)(sin x + 1) - Answer-cos x
/.Simplify the expression sin(x) + cos(x)cot(x) - Answer-csc(x)
/.Given a data set, use a graphing calculator or other device to determine the line of
best fit (least-squares regression line).
Let x high temperature (in degrees F) and y the number of cups of lemonade sold at a
local stand.
https://files.catbox.moe/sinjrh.PNG
Use the best fit line to predict the number of cups of lemonade that will sell when the
high temperature is 98. - Answer-57
/.If lim x→a f(x) exists, then f(a) exists. - Answer-False
/.True or False:
If you know in advance that lim x→4 f(x) exists, you can determine the value of the limit
by knowing all the values of f(x) for all x > 4? - Answer-True
/.lim x→0 (1/(x+3) - 1/3)/x
Added image b/c it's hard to read
https://files.catbox.moe/kbl61f.PNG - Answer--1/9
/.lim x→2- {2x-3, x<2 {x²+1, x≥2
piecewise: https://files.catbox.moe/ua2frr.PNG - Answer-1
/.Use the graph of f(x) above to determine which one of the following statements is true.
https://files.catbox.moe/4s8uqk.PNG - Answer-lim x→1 f(x) exists.
/.Use the graph of f(x) to evaluate the limit. lim x→2 f(x) =
https://files.catbox.moe/3tqjma.PNG - Answer-1
/.lim x→0+ |x|/x =
This is the greatest integer function. - Answer-0
/.lim x→0- 1/x^2 - Answer-+∞
/.lim x→∞ tan⁻¹(x²-x⁴) = - Answer--π/2
/.Find any Horizontal Asymptotes of
g(x) = (x-3)/(x2-4) - Answer-y=0
/.Evaluate the limit:
lim x→∞ (5x³+10x²-2)/(8x⁴+1) - Answer-0
