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WorksheetsCalculus Quiz
Total questions: 114
Worksheet time: 2hrs 54mins
What is the absolute maximum value of the function f(x) = x³ - 3x² + 2 on the interval [-1, 3]?
2
4
6
8
Which of the following is a critical point of the function f(x) = x⁴ - 8x² + 2?
x = 0
x = 2
x = -2
All of the above
On what interval is the function f(x) = x³ + 3x² - 9x + 5 increasing?
(-∞, -3)
(-3, 1)
(1, ∞)
(-∞, -3) U (1, ∞)
On what interval is the function f(x) = x⁴ - 4x³ + 6x² - 4x + 1 concave up?
(-∞, 1)
(1, ∞)
(-∞, ∞)
None of the above
The Mean Value Theorem for a function f(x) on the interval [a, b] states:
There exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a)
f(x) is continuous on [a, b]
f(x) is differentiable on (a, b)
All of the above
Which of the following conditions is necessary to apply L'Hopital's Rule?
The limit must be in an indeterminate form
The function must be continuous
The function must be differentiable
a and c
What is the antiderivative of the function f(x) = 2x?
x² + C
x²/2 + C
2x + C
2
Which of the following is the equation of the horizontal asymptote of the graph of f(x) = (3x² - 2x + 1)/(x² + 1)?
y = 0
y = 1
y = 3
There is no horizontal asymptote
Which of the following functions has an inflection point at x = 0?
f(x) = x³
f(x) = x⁴
f(x) = x²
f(x) = x
What is the derivative of the function f(x) = sin(x²)?
2x cos(x²)
cos(x²)
-2x cos(x²)
-cos(x²)
What is the derivative of the function f(x) = ln(x² + 1)?
1/(x² + 1)
2x/(x² + 1)
x/(x² + 1)
2/(x² + 1)
What is the indefinite integral of the function f(x) = e^(2x)?
e^(2x) + C
2e^(2x) + C
(1/2)e^(2x) + C
-e^(2x) + C
Which of the following is the equation of the vertical asymptote of the graph of f(x) = 1/(x - 1)?
x = 1
x = -1
y = 0
There is no vertical asymptote
What is the derivative of the function f(x) = tan(3x)?
sec²(3x)
3sec²(3x)
sec²(3x)/3
-3sec²(3x)
Which of the following functions has a local minimum value at x = 0?
f(x) = x²
f(x) = -x²
f(x) = x³
f(x) = -x³
Find all critical points of f(x) = x^4 - 2x^2.
x = -1, 0, 1
x = -1, 1
x = 0
x = -√2, 0, √2
The maximum value of the function f(x) = x^3 - 3x + 1 is:
1
2
3
4
Determine the intervals where f(x) = x^3 - 3x^2 + 2 is decreasing.
(-∞, 0) U (2, ∞)
(0, 2)
(-∞, ∞)
None
On which interval is the function f(x) = 1/x increasing?
(-∞, 0) U (0, ∞)
(-∞, 0)
(0, ∞)
None
Find all vertical asymptotes of f(x) = (x^2 + 4)/(x - 2).
x = 2
x = -2
x = -2, 2
No asymptotes
Find all horizontal asymptotes of f(x) = (2x + 1)
Find all vertical asymptotes of f(x) = (x^2 + 4)/(x - 2).
x = 2
x = -2
x = -2, 2
No asymptotes
Find all horizontal asymptotes of f(x) = (2x + 1)/(x - 1).
y = 2
y = 1
y = -1
No asymptotes
Rolle's Theorem is used to prove:
The Mean Value Theorem
The Intermediate Value Theorem
The Squeeze Theorem
None of the above
Which of the following values of c satisfies the Mean Value Theorem for f(x) = x^2 on the interval [0, 2]?
c = 1
c = 2
c = 0
No such value of c
On what interval is f(x) = x^3 - 12x concave up?
(-∞, 0)
(0, ∞)
(-∞, ∞)
None
What is the antiderivative of f(x) = sin(x)?
cos(x) + C
-cos(x) + C
sin(x) + C
-sin(x) + C
What is the antiderivative of f(x) = e^x?
e^x + C
ln(x) + C
x + C
No antiderivative
What is the indefinite integral of 1/x w.r.t. x?
ln|x| + C
-1/x^2 + C
x + C
No antiderivative
Find the derivative of f(x) = cos(2x).
-sin(2x)
-2sin(2x)
sin(2x)
2sin(2x)
Find the derivative of f(x) = e^(x^2).
e^(x^2)
2xe^(x^2)
x^2e^(x^2)
2e^(x^2)
Find the derivative of f(x) = ln(2x).
1/(2x)
1/x
2/x
1/(xln2)
What is the derivative of f(x) = sin(x)?
cos(x)
-cos(x)
sin(x)
-sin(x)
What is the derivative of f(x) = x²?
2x
x
x³/3
1/x
What is the derivative of f(x) = e^x?
e^x
ln(x)
x
1/x
What is the derivative of f(x) = ln(x)?
1/x
x
e^x
-1/x²
Which of the following functions has a local maximum value at x = 0?
f(x) = x²
f(x) = -x²
f(x) = x³
f(x) = -x³
Which of the following functions has a local minimum value at x = 0?
f(x) = x²
f(x) = -x²
f(x) = x³
f(x) = -x³
On what interval is the function f(x) = x³ - 3x increasing?
(-∞, -1) U (1, ∞)
(-1, 1)
(-∞, ∞)
None
On what interval is the function f(x) = x³ - 3x decreasing?
(-∞, -1) U (1, ∞)
(-1, 1)
(-∞, ∞)
None
On what interval is the function f(x) = x⁴ - 4x² concave up?
(-∞, -√(2/3)) U (√(2/3), ∞)
(-√(2/3), √(2/3))
(-∞, ∞)
None
On what interval is the function f(x) = x⁴ - 4x² concave down?
(-∞, -√(2/3)) U (√(2/3), ∞)
(-√(2/3), √(2/3))
(-∞, ∞)
None
What is the vertical asymptote of the graph of f(x) = (x - 1)/(x + 1)?
x = -1
x = 1
y = 1
No asymptote
What is the horizontal asymptote of the graph of f(x) = (x - 1)/(x + 1)?
x = -1
x = 1
y = 1
No asymptote
Find the x-coordinate of the inflection point of f(x) = x³ - 3x² + 2x + 1.
x = 1
x = 0
x = -1
No inflection point
Find the x-coordinate of the inflection point of f(x) = x³ - 3x² + 2x + 1.
x = 1
x = 0
x = -1
No inflection point
What is the derivative of the function f(x) = sinh(x)?
cosh(x)
-cosh(x)
sinh(x)
-sinh(x)
What is the derivative of the function f(x) = cosh(x)?
sinh(x)
-sinh(x)
cosh(x)
-cosh(x)
What is the Intermediate Value Theorem?
If f is continuous on [a, b] and k is between f(a) and f(b), then there is a c in (a, b) such that f(c) = k.
If f is differentiable on (a, b) and f(a) = f(b), then there is a c in (a, b) such that f'(c) = 0.
If lim f(x) = L and lim g(x) = M, then lim (f(x) + g(x)) = L + M.
If f is continuous on [a, b], then f attains both a maximum and a minimum value on [a, b].
Define a critical point of a function.
A point where the derivative is zero.
A point where the function is undefined.
A point where the derivative is zero or undefined.
A point where the function is discontinuous.
What is a vertical asymptote?
A horizontal line that the graph approaches as x approaches infinity.
A vertical line that the graph approaches as x approaches a specific value.
A slanted line that the graph approaches as x approaches infinity.
A point where the function is undefined.
What is a horizontal asymptote?
A vertical line the graph approaches as x approaches a specific value.
A horizontal line the graph approaches as x approaches infinity.
A slanted line the graph approaches as x approaches infinity.
A point where the derivative is zero.
What conditions must be met for a function to be continuous at a point?
The function is defined at that point.
The limit of the function exists at that point.
The limit of the function equals the function's value at that point.
All of the above.
What is an inflection point?
A point where the function changes from increasing to decreasing.
A point where the function changes from concave up to concave down or vice versa.
A point where the derivative is zero.
A point where the function is undefined.
Find the derivative of f(x) = 3x² - 5x + 2.
6x - 5
3x - 5
6x² - 5
6x² - 5x
Find the derivative of g(x) = sin(2x).
cos(2x)
2cos(2x)
-2cos(2x)
2sin(2x)
Find the derivative of h(x) = e^(3x).
e^(3x)
3e^(3x)
e^(3)
3e^(x)
Find the derivative of i(x) = ln(x² + 1).
1/(x² + 1)
2x/(x² + 1)
x/(x² + 1)
2/(x² + 1)
Evaluate the limit: lim (x² - 9) / (x - 3) as x approaches 3.
0
6
∞
undefined
Evaluate the limit: lim (sin x) / x as x approaches 0.
0
1
∞
undefined
Evaluate the limit: lim (1 - cos x) / x² as x approaches 0.
Evaluate the limit: lim (sin x) / x as x approaches 0.
0
1
∞
undefined
Evaluate the limit: lim (1 - cos x) / x² as x approaches 0.
0
1/2
1
∞
Evaluate the limit: lim x * ln(x) as x approaches 0 from the right.
0
1
∞
-∞
Find the limit: lim (x³ + 2x - 1) / (x² + 1) as x approaches 2.
11/5
1
0
undefined
Find the limit: lim (x² - 4) / (x - 2) as x approaches 2.
0
4
∞
undefined
Find the antiderivative of f(x) = 4x³.
12x² + C
x⁴ + C
x⁴/4 + C
4x⁴ + C
Find the antiderivative of f(x) = cos(x).
sin(x) + C
-sin(x) + C
-cos(x) + C
cos(x) + C
Find the antiderivative of f(x) = e^(2x).
e^(2x) + C
2e^(2x) + C
e^(2x)/2 + C
e^x + C
Find the antiderivative of f(x) = 1/x (x>0).
1/x² + C
ln|x| + C
x + C
-1/x² + C
Find the value of ∫₀¹ (2x + 1) dx.
1
2
3
4
Find the value of ∫₀² (x²) dx.
2/3
8/3
4
2
If f'(x) = 6x + 2, and f(1) = 5, find f(x).
3x² + 2x + 4
3x² + 2x
3x² + 2x + 0
6x + 2
Find the value of ∫₁² (1/x) dx.
0
ln 2
1
2
Find the area under the curve y = x from x=0 to x=2.
1
2
3
4
Find the area under the curve y = x² from x=0 to x=1.
1/2
1/3
1
2
Find the derivative of f(x) = x^4 - 6x² + 3x - 7.
4x³-12x+3
4x³-12x-7
4x²-12x+3
4x³+3
If f'(x) = 2x - 3 and f(0) = 4, what is f(2)?
2
0
-2
4
Evaluate lim (x^3 - 8)/(x - 2) as x approaches 2.
0
12
∞
undefined
Find the derivative of f(x) = (x+2)/(x-1).
2/(x-1)²
-3/(x-1)²
3/(x-1)²
-2/(x-1)²
Find the derivative of f(x) = 7x³ - 4x² + 5x - 9.
21x² - 8x + 5
7x² - 4x + 5
21x² - 4x + 5
7x³ - 4x² + 5
Find the derivative of g(x) = 2sin(x) + 3cos(x).
2cos(x) - 3sin(x)
2cos(x) + 3sin(x)
-2cos(x) + 3sin(x)
-2cos(x) - 3sin(x)
Find the derivative of h(x) = e^(-x²).
e^(-x²)
-2xe^(-x²)
-2e^(-x²)
2xe^(-x²)
Find the derivative of i(x) = ln(x³ + 1).
3x²/(x³ + 1)
1/(x³ + 1)
3x²
1/x³
Find the derivative of j(x) = (x² + 1)(x - 3).
3x² - 6x + 1
3x² - 6x -1
3x² + 1
x² - 6x + 1
Find the derivative of k(x) = (x + 2)/(x - 1).
-3/(x - 1)²
3/(x - 1)²
-3/(x + 2)²
3/(x + 2)²
Find the derivative of l(x) = √(x² + 4).
1/(2√(x² + 4))
x/(√(x² + 4))
x/2√(x²+4)
2x/(√(x² + 4))
Find the derivative of m(x) = (x³ + 2x)^(1/2).
(3x² + 2)/(2√(x³ + 2x))
(3x² + 2)√(x³ + 2x)
(3x² + 2)/√(x³ + 2x)
√(3x² + 2)/(x³ + 2x)
Find the derivative of n(x) = sin(e^x).
cos(e^x)e^x
cos(e^x)
-cos(e^x)e^x
e^xcos(e^x)
Find the derivative of o(x) = e^(sin x).
(a)
Find the derivative of n(x) = sin(e^x).
cos(e^x)e^x
cos(e^x)
-cos(e^x)e^x
e^xcos(e^x)
Find the derivative of o(x) = e^(sin x).
e^(sin x)cos(x)
e^(cos x)
e^(sin x)
-e^(sin x)cos(x)
Find dy/dx if x² + y² = 25. (Assume y is a function of x).
-x/y
x/y
y/x
-y/x
Find dy/dx if x³ + y³ = 3xy. (Assume y is a function of x).
(y - x²)/(y² - x)
(x - y²)/(x² - y)
(y - x²)/(y² - x)
(y² - x)/(y - x²)
Find dy/dx if x = cos(t) and y = sin(t).
tan(t)
-tan(t)
cot(t)
-cot(t)
Find dy/dx if y = x⁴ - 2x² + 5, at x = 1.
0
-2
2
4
Evaluate the limit lim (x² - 16) / (x - 4) as x → 4.
0
8
undefined
16
Evaluate the limit lim (sin(3x)) / x as x → 0.
0
1
3
undefined
Evaluate the limit lim (e^x - 1) / x as x → 0.
0
1
e
undefined
Evaluate the limit lim (1 - cos x) / x as x → 0.
0
1
-1
undefined
Evaluate the limit lim (x² + 3x - 10) / (x - 2) as x → 2.
0
7
undefined
10
Evaluate the limit lim (x - 1) / (√x - 1) as x → 1.
0
1
2
undefined
Find the equation of the tangent line to y = x² - 2x at x = 3.
y = 4x - 3
y = 4x - 9
y = 3x - 3
y = 3x - 6
Find the equation of the tangent line to y = e^x at x = 0.
y = x + 1
y = x
y = x -1
y = e^x
If f'(x) = 4x + 1 and f(0) = 2, find f(2).
2
6
10
12
If f'(x) = cos(x) and f(0) = 1, find f(π/2).
0
1
2
π/2
A particle moves along the x-axis such that its position at time t is given by x(t) = t³ - 6t² + 9t. Find its velocity at t = 2.
3
0
-3
9
The radius of a circle is increasing at a rate of 2 cm/s. How fast is the area increasing when the radius is 5 cm? (Area of a circle = πr²)
10π cm²/s
20π cm²/s
2π cm²/s
5π cm²/s
A ladder 10 feet long leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?
-3 ft/s
3 ft/s
-2 ft/s
2 ft/s
Evaluate ∫ (x³ + 2x) dx
x⁴ + x² + C
x⁴/4 + x² + C
3x² + 2 + C
x⁴ + 2x² + C
Evaluate ∫ (e^x - 1) dx
e^x - x + C
e^x - 1 + C
e^x + x + C
e^x + C
Evaluate ∫ cos(2x) dx
sin(2x) + C
2sin(2x) + C
sin(2x)/2 + C
-sin(2x)/2 + C
Evaluate ∫ (1/x) dx (x > 0)
1/x² + C
x + C
ln|x| + C
-1/x² + C
Evaluate ∫₀¹ (x² + 1) dx
1/3
4/3
2
1
