Differentiate. y = x + 1 / (x + 1)
> Figure 22 gives the number of U.S. farms in millions from 1920 (t = 20) to 2000 (t = 100). In what year was the number of farms decreasing most rapidly? Figure 22: U.S. Farms (millions) 52428766 Y HT 0 10 20 30 40 50 60 70 80 90 100 (1900) (2000) T
> Suppose that some organic waste products are dumped into a lake at time t = 0 and that the oxygen content of the lake at time t is given by the graph in Fig. 21. Describe the graph in physical terms. Indicate the significance of the inflection point at t
> One method of determining the level of blood flow through the brain requires the person to inhale air containing a fixed concentration of N2O, nitrous oxide. During the first minute, the concentration of N2O in the jugular vein grows at an increasing rat
> Let C(x) denote the total cost of manufacturing x units of some product. Then C(x) is an increasing function for all x. For small values of x, the rate of increase of C(x) decreases (because of the savings that are possible with “mass production”). Event
> In certain professions, the average annual income has been rising at an increasing rate. Let f (T) denote the average annual income at year T for persons in one of these professions and sketch a graph that could represent f (T).
> Differentiate. y = 3x + π3
> In certain professions, the average annual income has been rising at an increasing rate. Let f (T) denote the average annual income at year T for persons in one of these professions and sketch a graph that could represent f (T).
> The annual world consumption of oil rises each year. Furthermore, the amount of the annual increase in oil consumption is also rising each year. Sketch a graph that could represent the annual world consumption of oil.
> Draw the graph of a function y = f (x) with the stated properties. Both the function and the slope decrease as x increases.
> Draw the graph of a function y = f (x) with the stated properties. The function decreases and the slope increases as x increases.
> Draw the graph of a function y = f (x) with the stated properties. The function increases and the slope decreases as x increases.
> Draw the graph of a function y = f (x) with the stated properties. Both the function and the slope increase as x increases.
> Refer to the graph in Fig. 20. Figure 20: (a) At which labeled points is the function decreasing? (b) At which labeled points is the graph concave down? (c) Which labeled point has the most negative slope (that is, negative and with the greatest magni
> Refer to the graph in Fig. 20. Figure 20: (a) At which labeled points is the function increasing? (b) At which labeled points is the graph concave up? (c) Which labeled point has the most positive slope? Figure 20 A B ایت از D E / y = f(x)
> Describe the way the slope changes on the graph in Exercise 10. Exercise 10: Describe the following graph.
> Describe the way the slope changes on the graph in Exercise 8. Exercise 8: Describe the following graph. " x
> Differentiate. y = 1 / x3 + x + 1
> Describe the way the slope changes on the graph in Exercise 6. Exercise 6: Describe the following graph. THE + Y
> Describe the way the slope changes as you move along the graph (from left to right) in Exercise 5. Exercise 5: Describe the following graph. 引
> Describe the following graph. Y A x
> Describe the following graph.
> Describe the following graph.
> Describe the following graph. 1 y=x
> Describe the following graph. " x
> Describe the following graph. 1
> Find the equation and sketch the graph of the following lines. Parallel to -2x + 3y = 6, passing through (0, 1).
> Find the equation and sketch the graph of the following lines. Parallel to y = -2x, passing through (3, 5).
> Differentiate. f(x) = 5√(3x3 + x)
> Find the equation and sketch the graph of the following lines. Through (1, 4), with slope - 13.
> Find the equation and sketch the graph of the following lines. Through (2, 0), with slope 5.
> Find the equation and sketch the graph of the following lines. With slope 3/4 , y-intercept (0, -1).
> Find the equation and sketch the graph of the following lines. With slope -2, y-intercept (0, 3).
> Use limits to compute the following derivatives. As h approaches 0, what value is approached by 1 1 2+h 2 h
> Use limits to compute the following derivatives. What geometric interpretation may be given to ((3 + h)2 – 32 )/h in connection with the graph of f (x) = x2?
> Use limits to compute the following derivatives. f ‘(3), where f (x) = x2 - 2x + 1.
> Use limits to compute the following derivatives. f ‘(5), where f (x) = 1/(2x).
> Determine whether the following limits exist. If so, compute the limit. lim x→5 (x – 5) / (x2 – 7x + 2)
> Determine whether the following limits exist. If so, compute the limit. lim x→4 (x – 4) / (x2 – 8x + 16)
> Differentiate. y = 2 4√(x2 + 1)
> Determine whether the following limits exist. If so, compute the limit. lim x→3 1 / (x2 – 4x + 3)
> Determine whether the following limits exist. If so, compute the limit. lim x→2 (x2 – 4) / (x – 2)
> If you deposit $100 in a savings account at the end of each month for 2 years, the balance will be a function f (r) of the interest rate, r%. At 7% interest (compounded monthly), f (7) = 2568.10 and f ‘(7) = 25.06. Approximately how much additional money
> Let h(t) be a boy’s height (in inches) after t years. If h’(12) = 1.5, how much will his height increase (approximately) between ages 12 and 12 ½ ?
> The number of people riding the subway daily from Silver Spring, Maryland, to Washington’s Metro Center is a function f (x) of the fare, x cents. If f (235) = 4600 and f ‘(235) = -100, approximate the daily number of riders for each of the following cost
> A manufacturer estimates that the hourly cost of producing x units of a product on an assembly line is C(x) = .1x3 - 6x2 + 136x + 200 dollars. (a) Compute C(21) - C(20), the extra cost of raising the production from 20 to 21 units. (b) Find the marginal
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: Without calculating velocities, determine whether the person is traveling faster at t = 5 or at t = 6. y 12 11 10 9
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: What is the person’s velocity at time t = 3? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: What is the person’s average velocity from time t = 1 to t = 4? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y
> Refer to Fig. 3, where s(t) is the number of feet traveled by a person after t seconds of walking along a straight path. Figure 3: How far has the person traveled after 6 seconds? y 12 11 10 9 00 8 7 6 5 4 3 2 1 y = s(t) 0 1 2 3 4 5 6 7 Figure 3 W
> Each day the total output of a coal mine after t hours of operation is approximately 40t + t2 – 1 /15 t3 tons, 0 ≤ t ≤ 12. What is the rate of output (in tons of coal per hour) at t = 5 hours?
> A helicopter is rising at a rate of 32 feet per second. At a height of 128 feet the pilot drops a pair of binoculars. After t seconds, the binoculars have height s(t) = -16t2 + 32t + 128 feet from the ground. How fast will they be falling when they hit t
> In Fig. 2, the straight line is tangent to the graph of f (x) = x3. Find the value of a. Figure 2: a Figure 2 (0, 2) Y
> In Fig. 1, the straight line has slope -1 and is tangent to the graph of f (x). Find f (2) and f ‘(2). Figure 1: m = -1 Figure 1 Y 2 5 y = f(x) x
> Determine the equation of the tangent line to the curve y = (2x2 - 3x)3 at x = 2.
> Determine the equation of the tangent line to the curve y = 3x3 - 5x2 + x + 3 at x = 1.
> Find the equation of the tangent line to the curve y = x2 at the point (-2, 4). Sketch the graph of y = x2 and sketch the tangent line at (-2, 4).
> Find the equation of the tangent line to the curve y = x2 at the point (3/2 , 9/4). Sketch the graph of y = x2 and sketch the tangent line at (3/2, 9/4).
> What is the slope of the curve y = 1/(3x - 5) at x = 1? Write the equation of the line tangent to this curve at x = 1.
> What is the slope of the graph of f (x) = x3 - 4x2 + 6 at x = 2? Write the equation of the line tangent to the graph of f (x) at x = 2.
> Differentiate. y = 2 / x + 1
> 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
