midterm review calculus. unit 0 page 3 determine whether is rational or irrational. determine...
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Midterm Review
Calculus
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UNIT 0
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Page 3
Determine whether is rational or irrational.
Determine whether the given value of x satisfies the inequality:
a.) x = -2 b.) x = 0 c.) x = d.) x = -6
RATIONAL
SATISFIES DOES NOTSATISFY
SATISFIES SATISFIES
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Page 4
1.) 2.)
3.) 4.)
Solve each inequality:
x ≥ 3 -1 < x < 7
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Page 5
Given the interval [-3, 7], find:a.) the distance between -3 and 7
b.) the midpoint of the interval
c.) Use absolute value to describe this interval
d = 10
Midpoint = 2
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Page 6
Simplify each:
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Page 7
Remove all possible factors from the radical:
Complete the factorization:
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Page 8
1.) 2.)
3.)
Factor each completely:
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Page 9
Use the rational zero theorem to find all real roots of:
Possible Rational Zeros: ±1, ±2, ±3, ±6
So -1, 2, and 3 are all roots
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Page 10
Combine terms and simplify each:
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Page 11
Combine terms and simplify each:
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Page 12
Rationalize the denominator:
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UNIT 1
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Page 14
Find the distance between (3, 7) and (4, -2)
Find the midpoint of the line segment joining (0, 5) and (2, 1)
Determine whether the points (0, -3) , (2, 5) , and (-3, -15) are collinear.
Midpoint (1, 3)
All points are collinear
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Page 15
Find x so that the distance between (0, 3) and (x, 5) is 7
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Page 16
Sketch the graph of each:
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Page 17
Write the equation of the circle in standard form and sketch it:
Find the points of intersection of the graphs of:
(0, -5) and (4, -3)
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Page 18
Find the general equation of the line given certain information:a.) (7, 4) and (6, -2)
b.) (-2, -1) and slope = ⅔
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Page 19
Find the general equation of the line given certain information:a.) (6, -8) and undefined slope
b.) (0, 3) and perpendicular to 2x – 5y = 7
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Page 20
f(3) f(-6)
f(x – 5) f(x + Δx)
Given find the following:
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Page 21
Find the domain and range of:
Given and find:
Domain: (-∞, 3]Range: [0, ∞)
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Page 22
Given find
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Page 23
1.) 2.)
3.) 4.)
Find each limit:
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Page 24
Find the
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Page 25
Find the discontinuities of each and tell which are removable.
x = ±8
x = 8 is removable
x = 3 is a non-removable discontinuity
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Page 26
Sketch the graph:
Hole @ x = 2
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UNIT 2
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Page 28
Find the derivative of each:1.)
2.)
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Page 29
Use the derivative to find the equation of the tangent line to the graph of f(x) at the point (6, 2)
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Page 30
Find f’(x) for each f(x)1.)
2.)
3.)
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Page 31
Find the average rate of change of f(x) over the interval [0. 2]. Compare this to the instantaneous rate of change at the endpoints of the interval.
Average rate of change: 4
Instantaneous rates of change:
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Page 32
Given the cost function C(x), find the marginal cost of producing x units.
Marginal cost: 4.31 – 0.0002x
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Page 33
Find f’(x) for each f(x)1.)
2.)
3.)
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Page 34
Find f’(x) for each f(x)1.)
2.)
3.)
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Page 35
Find the derivative of each:1.)
2.)
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Page 36
1.) Given f(x), find f’’’(x)
2.) Given f(x), find f’’’’(x)
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Page 37
Use implicit differentiation to find
1.)
2.)
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Page 38
Use implicit differentiation to find
1.)
2.)
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Page 39
Let y = 3x2 . Find when x = 2 and = 5
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Page 40
The area A of a circle is increasing at a rate of 10 in.2/min. Find the rate of change of the radius r when r = 4 inches.
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Page 41
The volume of a cone is .
Find the rate of change of the height when :
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UNIT 3.1-3.4
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Page 43
Find the critical numbers and the intervals on which f(x) is increasing or decreasing for f(x):
Increasing: (-∞, 0) U (4, ∞)Decreasing: (0, 4)
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Page 44
Find the critical numbers and the intervals on which f(x) is increasing or decreasing for f(x):
Increasing: (-∞, ⅔) Decreasing: (⅔, 1)
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Page 45
Find the relative extrema of f(x)
Relative Minimum: (2, -45)
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Page 46
Find the relative extrema of f(x)
Relative Minimum: (-3, 0)
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Page 47
Find the absolute extrema of f(x) on [0, 5]
Abs. Max: (5, 0)Abs. Min: (2, -9)
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Page 48
Find the points of inflection of f(x)
No Inflections Points
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Page 49
Find the points of inflection of f(x)
Points of Inflection:
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Page 50
Find two positive numbers who product is 200 such that the sum of the first plus three times the second is a minimum.
First number:
Second number:
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Page 51
Three rectangular fields are to be enclosed by 3000 feet of fencing, as shown below. What dimensions should be used so that the enclosed area will be a maximum?
y
x x x
3x = 750 feet, y = 375 feet