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Adamson U
Calculus
Practice Quiz (with Answer)

Questions:

1
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Find the derivative of y = x^2
3
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Find the second derivative of y = cos(x)
5
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Find the equation of the tangent line to the graph of y = x^3 at the point (1,1)
7
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Find the equation of the normal line to the graph of y = 2x^2 at the point (1,2)
9
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Find the derivative of y = ln(x)
11
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Find the equation of the tangent line to the graph of y = sin(x) at the point
(π/2,1)
13
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Find the equation of the normal line to the graph of y = x^3 at the point (2,8)
15
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Find the derivative of y = e^x
17
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Find the equation of the tangent line to the graph of y = ln(x) at the point (1,0)
19
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Find the equation of the normal line to the graph of y = sin(x) at the point
(π/4,1/√2)

Answer:

1
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The derivative of sin(x) is cos(x) and the derivative of x is 1
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Find the derivative of y = x^2:
The derivative of y = x^2 is dy/dx = 2x

3
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So the definite integral is:
(x^3)/3 from 0 to 1 = (1^3)/3 - (0^3)/3 = (1/3) - 0 = 1/3

4
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Evaluate the indefinite integral of (1/x) dx:
The antiderivative of (1/x) is ln|x| + C, where C is the constant of integration
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Find the equation of the tangent line to the graph of y = x^3 at the point (1,1):
We can use the point-slope form of a line to find the equation of the tangent line
...
The point
(1,1) lies on the tangent line, so we can substitute these values into the pointslope form of a line:
y - 1 = 3(x - 1)

7
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So the definite integral is:
e^x from 0 to 1 = e^1 - e^0 = e - 1

8
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The slope of the normal line is the negative reciprocal of the derivative of y =
2x^2, which is 4x
...
Evaluate the limit as x approaches infinity of (1+1/x)^x:
This limit is equal to e^(1), where e is the base of the natural logarithm
...
Find the derivative of y = ln(x):
The derivative of ln(x) is 1/x
11
...
Find the equation of the tangent line to the graph of y = sin(x) at the point
(π/2,1):
We can use the point-slope form of a line to find the equation of the tangent line
...
The
point (π/2,1) lies on the tangent line, so we can substitute these values into the
point-slope form of a line:
y - 1 = cos(π/2)(x - π/2)

13
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Find the equation of the normal line to the graph of y = x^3 at the point (2,8):

We can use the point-slope form of a line to find the equation of the normal line
...
The point (2,8) lies on the normal line, so we can substitute these
values into the point-slope form of a line:
y - 8 = -(1/3)(x - 2)

15
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16
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Evaluate the definite integral of x^3 dx from x=1 to x=2:
The antiderivative of x^3 is (x^4)/4, so the definite integral is:
(x^4)/4 from 1 to 2 = (2^4)/4 - (1^4)/4 = (16/4) - (1/4) = 3/4

18
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The slope of the tangent line is the derivative of y = ln(x), which is 1/x
...
Evaluate the definite integral of (1/x^3) dx from x=1 to x=2:
The antiderivative of (1/x^3) is -1/x^2, so the definite integral is:
-1/x^2 from 1 to 2 = (-1/2^2) - (-1/1^2) = (-1/4) - (-1) = 3/4

20
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The slope of the normal line is the negative reciprocal of the derivative of y =
sin(x), which is cos(x)

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