Find the equation and sketch the graph of the following lines. The x-axis.
> Compute. d/dt (dy/dt), where y =1/3t.
> Compute. d2y/dx2, where y = 4x3/2
> Compute. d2/dP2 (3P + 2) |P=4
> Compute. d2/dt2 (t3 + 2t2 - t) |t= -1
> Compute. d2/dt2 (2√t)
> Compute. d2dx2 (5x + 1)4
> Compute. d dx (4x - 10)5 |x=3
> Compute. d/dz (z3 - 4z2 + z - 3) |z= -2
> Compute. d/dn (n-5)
> Compute. d/dP (√(1 - 3P))
> Differentiate. y = 1 / x3 + 1
> Compute. d/dt (t5/2 + 2t3/2 - t1/2)
> Compute. d/dx (x4 - 2x2)
> Find the slope of the graph of y = (4 - x)5 at x = 5.
> Find the slope of the graph of y = (3x - 1)3 - 4(3x - 1)2 at x = 0.
> If g(t) = ¼ (2t - 7)4, what is g’(3)?
> If f (x) = x5/2, what is f ‘(4)?
> If h(x) = - 1/2, find h(-2) and h’(-2).
> If g(u) = 3u - 1, find g(5) and g’(5).
> If V(r) = 15πr2, find V’(1/3 ).
> If f (t) = 3t3 - 2t2, find f ‘(2).
> Differentiate. y= (x2 + 1)2 + 3(x2 - 1)2
> Differentiate. y = 3x4
> Differentiate. f (x) = √(x + √x)
> Differentiate. h(x) = 32 x3/2 - 6x2/3
> Differentiate. g(P) = 4P0.7
> Differentiate. f (t) = 2 / t - 3t3
> Differentiate. h(t) = 3√2
> Differentiate. g(t) = 3√t – 3/√t
> Differentiate. f (t) = t10 - 10t9
> Differentiate. f (x) = [x5 - (x - 1)5]10
> Differentiate. f (x) = 5x/2 – 2/5x
> Differentiate. f (x) = 5
> Differentiate. y = 1 / 5x5
> Differentiate. f (x) = (2x + 1)3
> Differentiate. f (x) = 1/4√x
> Differentiate. y = 5 / 7x2 + 1
> Differentiate. y = √(x2 + 1)
> Differentiate. y = (x3 + x2 + 1)5
> Differentiate. y = 1 / 5x - 1
> Differentiate. y = ¾ x4/3 + 4/3 x3/4
> Differentiate. y = (3x2 - 1)8
> Differentiate. y = x4 – 4/x
> Differentiate. y = 3/x
> Differentiate. y = (x - 1)3 + (x + 2)4
> Differentiate. y = x7 + 3x5 + 1
> Differentiate. y = 6√x
> Differentiate. y = 5x8
> Differentiate. y = x7 + x3
> Find the equation and sketch the graph of the following lines. The y-axis.
> Find the equation and sketch the graph of the following lines. Vertical and 4 units to the right of the y-axis.
> Find the equation and sketch the graph of the following lines. Horizontal with height 3 units above the x–axis.
> Find the equation and sketch the graph of the following lines. Perpendicular to 3x + 4y = 5, passing through (6, 7).
> Find the equation and sketch the graph of the following lines. Perpendicular to y = 3x + 4, passing through (1, 2).
> Differentiate. y = 2x + (x + 2)3
> Find the equation and sketch the graph of the following lines. Through (2, 1) and (5, 1).
> Find the equation and sketch the graph of the following lines. Through (-1, 4) and (3, 7).
> What can you say about the slopes of parallel lines? Perpendicular lines?
> Describe how to find an equation for a line when you know the coordinates of two points on the graph of a line.
> What is the point–slope form of the equation of a line?
> Define the slope of a nonvertical line and give a physical description.
> How do you determine the proper units for a rate of change? Give an example.
> Describe marginal cost in your own words.
> What expression involving a derivative gives an approximation to f (a + h) - f (a)?
> Explain the relationship between derivatives and velocity and acceleration.
> Differentiate. y = 2√(x + 1)
> How is an (instantaneous) rate of change related to average rates of change?
> What is meant by the average rate of change of a function over an interval?
> Give two different notations for the first derivative of f (x) at x = 2. Do the same for the second derivative.
> State the general power rule and give an example.
> In your own words, explain the meaning of “ f (x) is differentiable at x = 2.” Give an example of a function f (x) that is not differentiable at x = 2.
> In your own words, explain the meaning of “ f (x) is continuous at x = 2.” Give an example of a function f (x) that is not continuous at x = 2.
> In your own words, explain the meaning of lim x→∞ f (x) = 3. Give an example of such a function f (x). Do the same for lim x→∞ f (x) = 3.
> Give the limit definition of f ‘(2), that is, the slope of f (x) at the point (2, f (2)).
> In your own words, explain the meaning of lim x→2 f (x) = 3. Give an example of a function with this property.
> Explain how to calculate f ‘(2) as the limit of slopes of secant lines through the point (2, f (2)).
> Differentiate. y = 3 3√(2x2 + 1)
> State the power rule, the constant-multiple rule, and the sum rule, and give an example of each.
> Explain why the derivative of a constant function is 0.
> What does f ‘(2) represent?
> Give a physical description of what is meant by the slope of f (x) at the point (2, f (2)).
> Suppose that f (x) = 4x2. (a) What is the average rate of change of f (x) over each of the intervals 1 to 2, 1 to 1.5, and 1 to 1.1? (b) What is the (instantaneous) rate of change of f (x) when x = 1?
> If f (x) = 3x2 + 2, calculate the average rate of change of f (x) over the following intervals (a) 0 ≤ x ≤ .5, (b) 0 ≤ x ≤ .1, (c) 0 ≤ x ≤ .0.01.
> If f (x) = x2 + 3x, calculate the average rate of change of f (x) over the following intervals (a) 1 …≤ x ≤ 2, (b) 1 ≤ x ≤ 1.5, (c) 1 ≤ x ≤ 1.1.
> A ball thrown straight up into the air has height s(t) = 102t - 16t2 feet after t seconds. (a) Display the graphs of s(t) and s’(t) in the window [0, 7] by [-100, 200]. Use these graphs to answer the remaining questions. (b) How high is the ball after 2
> In a psychology experiment, people improved their ability to recognize common verbal and semantic information with practice. Their judgment time after t days of practice was f (t) = .36 + .77(t - .5)-0.36 seconds. (a) Display the graphs of f (t) and f ‘(
> In an 8-second test run, a vehicle accelerates for several seconds and then decelerates. The function s(t) gives the number of feet traveled after t seconds and is graphed in Fig. 9. (a) How far has the vehicle traveled after 3.5 seconds? (b) What is the
> Differentiate. y = 4(x2 - 6)-3
> National health expenditures (in billions of dollars) from 1980 to 1998 are given by the function f (t) in Fig. 8. (a) How much money was spent in 1987? (b) Approximately how fast were expenditures rising in 1987? (c) When did expenditures reach $1 trill
> Estimate how much the function f (x) = 1 / 1 + x2 will change if x decreases from 1 to .9.
> Consider the cost function C(x) = 6x2 + 14x + 18 (thousand dollars). (a) What is the marginal cost at production level x = 5? (b) Estimate the cost of raising the production level from x = 5 to x = 5.25. (c) Let R(x) = -x2 + 37x + 38 denote the revenue i
> Let f (x) be the value in dollars of one share of a company x days since the company went public. (a) Interpret the statements f (100) = 16 and f ‘(100) = .25. (b) Estimate the value of one share on the 101st day since the company went public.
> Let P(x) be the profit (in dollars) from manufacturing and selling x cars. Interpret P(100) = 90,000 and P’(100) = 1200. Estimate the profit from manufacturing and selling 99 cars.
> Let C(x) be the cost (in dollars) of manufacturing x items. Interpret the statements C(2000) = 50,000 and C’(2000) = 10. Estimate the cost of manufacturing 1998 items.
> Let f (x) be the number (in thousands) of computers sold when the price is x hundred dollars per computer. Interpret the statements f (12) = 60 and f ‘(12) = -2. Then, estimate the number of computers sold if the price is set at $1250 per computer.
> Let f (x) be the number of toys sold when x dollars are spent on advertising. Interpret the statements f (100,000) = 3,000,000 and f ‘(100,000) = 30.
> Sales Let f (p) be the number of cars sold when the price is p dollars per car. Interpret the statements f (10,000) = 200,000 and f ‘(10,000) = -3.
> Suppose that 5 mg of a drug is injected into the bloodstream. Let f (t) be the amount present in the bloodstream after t hours. Interpret f (3) = 2 and f ‘(3) = -.5. Estimate the number of milligrams of the drug in the bloodstream after 312 hours.
> Differentiate. y = 4/x2
> Let f (t) be the temperature of a cup of coffee t minutes after it has been poured. Interpret f (4) = 120 and f ‘(4) = -5. Estimate the temperature of the coffee after 4 minutes and 6 seconds, that is, after 4.1 minutes.
