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 = -2b.) x = 0c.) x = d.) x = -6 RATIONAL SATISFIESDOES NOT SATISFY 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: 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 – x

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 = 3x 2. 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

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