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Midterm Review
Calculus
UNIT 0
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
Page 4
1.) 2.)
3.) 4.)
Solve each inequality:
x ≥ 3 -1 < x < 7
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
Page 6
Simplify each:
Page 7
Remove all possible factors from the radical:
Complete the factorization:
Page 8
1.) 2.)
3.)
Factor each completely:
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
Page 10
Combine terms and simplify each:
Page 11
Combine terms and simplify each:
Page 12
Rationalize the denominator:
UNIT 1
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
Page 15
Find x so that the distance between (0, 3) and (x, 5) is 7
Page 16
Sketch the graph of each:
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)
Page 18
Find the general equation of the line given certain information:a.) (7, 4) and (6, -2)
b.) (-2, -1) and slope = ⅔
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
Page 20
f(3) f(-6)
f(x – 5) f(x + Δx)
Given find the following:
Page 21
Find the domain and range of:
Given and find:
Domain: (-∞, 3]Range: [0, ∞)
Page 22
Given find
Page 23
1.) 2.)
3.) 4.)
Find each limit:
Page 24
Find the
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
Page 26
Sketch the graph:
Hole @ x = 2
UNIT 2
Page 28
Find the derivative of each:1.)
2.)
Page 29
Use the derivative to find the equation of the tangent line to the graph of f(x) at the point (6, 2)
Page 30
Find f’(x) for each f(x)1.)
2.)
3.)
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:
Page 32
Given the cost function C(x), find the marginal cost of producing x units.
Marginal cost: 4.31 – 0.0002x
Page 33
Find f’(x) for each f(x)1.)
2.)
3.)
Page 34
Find f’(x) for each f(x)1.)
2.)
3.)
Page 35
Find the derivative of each:1.)
2.)
Page 36
1.) Given f(x), find f’’’(x)
2.) Given f(x), find f’’’’(x)
Page 37
Use implicit differentiation to find
1.)
2.)
Page 38
Use implicit differentiation to find
1.)
2.)
Page 39
Let y = 3x2 . Find when x = 2 and = 5
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.
Page 41
The volume of a cone is .
Find the rate of change of the height when :
UNIT 3.1-3.4
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)
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)
Page 45
Find the relative extrema of f(x)
Relative Minimum: (2, -45)
Page 46
Find the relative extrema of f(x)
Relative Minimum: (-3, 0)
Page 47
Find the absolute extrema of f(x) on [0, 5]
Abs. Max: (5, 0)Abs. Min: (2, -9)
Page 48
Find the points of inflection of f(x)
No Inflections Points
Page 49
Find the points of inflection of f(x)
Points of Inflection:
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:
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
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