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mathematics
applied calculus

Questions and Answers of Applied Calculus

Find the area under P = 100(0.6)t between t = 0 and t = 8.
Explain in words what the integral represents and give units.∫60 a(t) dt, where a(t) is acceleration in km/hr2 and t is time in hours.
Find the average value of the function over the given interval. g(t) = 1 + t for 0 ≤ t ≤ 2
The marginal cost of drilling an oil well depends on the depth at which you are drilling; drilling becomes more expensive, per meter, as you dig deeper into the earth. The fixed costs are 1,000,000
Figure 5.10 shows the velocity of an object for 0 ≤ t ≤ 8. Calculate the following estimates of the distance the object travels between t = 0 and t = 8, and indicate whether each is an upper or
Find the total area between y = 4 − x2 and the x-axis for 0 ≤ x ≤ 3.
Explain in words what the integral represents and give units.∫20112005 f(t) dt, where f(t) is the rate at which world population is growing in year t, in billion people per year.
Estimate ∫60 2x dx using a left-hand sum with n = 2.
Find the average value of the function over the given interval.g(t) = et for 0 ≤ t ≤ 10
The population of Tokyo grew at the rate shown in Figure 5.74. Estimate the change in population between 1970 and 1990. rate of growth (million/year) 0.5 1970 Figure 5.74 P' (t) 1990 t (years)
Find the area between y = x + 5 and y = 2x + 1 between x = 0 and x = 2.
Explain in words what the integral represents and give units.∫50 s(x) dx, where s(x) is rate of change of salinity (salt concentration) in gm/liter per cm in sea water, and where x is depth below
Figure 5.11 shows the velocity of a car for 0 ≤ t ≤ 12 and the rectangles used to estimate the distance traveled.(a) Do the rectangles represent a left or a right sum?(b) Do the rectangles lead
Estimate ∫120 1/x + 1 dx using a left-hand sum with n = 3.
Estimate ∫10 e−x2 dx using n = 5 rectangles to form a(a) Left-hand sum (b) Right-hand sum
Estimate the average value of the function between x = 0 and x = 7. 42 19 て. 2 5 f(x) 6 X
Figure 5.76 shows the rate of change of the quantity of water in a water tower, in liters per day, during the month of April. If the tower had 12,000 liters of water in it on April 1, estimate the
(a) What is the total area between the graph of f(x) in Figure 5.42 and the x-axis, between x = 0 and x = 5?(b) What is ∫50 f(x) dx? Area = 6 f(x) 3 Area = 7 Figure 5.42
Oil leaks out of a tanker at a rate of r = f(t) gallons per minute, where t is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first
Figure 5.13 shows the velocity of a car for 0 ≤ t ≤ 24 and the rectangles used to estimate the distance traveled.(a) Do the rectangles represent a left or a right sum?(b) Do the rectangles lead
The concentration of a medication in the plasma changes at a rate of ℎ(t) mg/ml per hour, t hours after the delivery of the drug.(a) Explain the meaning of the statement ℎ(1) = 50.(b) There is
(a) What is the average value of f(x) = √1 − x2 over the interval 0 ≤ x ≤ 1?(b) How can you tell whether this average value is more or less than 0.5 without doing any calculations?
Using Figure 5.43, decide whether each of the following definite integrals is positive or negative.(a) ∫−4−5 f(x) dx (b) ∫1−4 f(x) dx (c) ∫31 f(x) dx (d) ∫3−5 f(x) dx
Pollution is removed from a lake at a rate of f(t) kg∕day on day t.(a) Explain the meaning of the statement f(12) = 500.(b) If ∫155 f(t) dt = 4000, give the units of the 5, the 15, and the
A car comes to a stop six seconds after the driver applies the brakes. While the brakes are on, the velocities recorded are in Table 5.5.(a) Give lower and upper estimates for the distance the car
A car starts moving at time t = 0 and goes faster and faster. Its velocity is shown in the following table. Estimate how far the car travels during the 12 seconds. t (seconds) 0 3 6 9 12 Velocity
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. -x² dx, n = 5 -24
Estimate the average value of f(x) from x = a to x = b. 2 1 1 a f(x) b x
The total cost in dollars to produce q units of a product is C(q). Fixed costs are $20,000. The marginal cost isC'(q) = 0.005q2 − q + 56.(a) On a graph of C'(q), illustrate graphically the total
Annual coal production in the US (in billion tons per year) is given in the table. Estimate the total amount of coal produced in the US between 1997 and 2009. If r = f(t) is the rate of coal
In Problem Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. -8 -1 2* dx, n = 3
Estimate the average value of f(x) from x = a to x = b. 20 15 10 in a f(x) b X
The marginal cost C'(q) (in dollars per unit) of producing q units is given in the following table.(a) If fixed cost is $10,000, estimate the total cost of producing 400 units.(b) How much would the
(a) Estimate (by counting the squares) the total area shaded in Figure 5.44.(b) Using Figure 5.44, estimate ∫80 f(x) dx.(c) Why are your answers to parts (a) and (b) different?
The table gives annual US emissions, H(t), of hydrofluorocarbons, or “super greenhouse gases,” in millions of metric tons of carbon-dioxide equivalent. Let t be in years since 2000.(a) What are
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. - 12 0 x² dx, n = 4
Estimate the average value of the function between x = 0 and x = 7. ∞ 6+2 9 4 JUN 1 2 3 4 5 6 7 f(x)
Figure 5.12 shows the velocity of a runner for 0 ≤ t ≤ 15 and the rectangles used to estimate the distance traveled.(a) Do the rectangles represent a left or a right sum?(b) Do the rectangles
Your velocity is v(t) = ln(t2+1) ft∕sec for t in seconds, 0 ≤ t ≤ 3. Find the distance traveled during this time.
A marginal cost function C'(q) is given in Figure 5.75. If the fixed costs are $10,000, estimate:(a) The total cost to produce 30 units.(b) The additional cost if the company increases production
Find the area enclosed by y = 3x and y = x2.
Refer to Fig. 10. Describe what happens to cos t as t increases from 0 to π. (a) P P (b) P P (d) Figure 10 Movement along the unit circle. (e) P (c)
Find t such that 0 ≤ t ≤ π and t satisfies the stated condition.cos t = cos(-5π/8)
Evaluate the following integrals using techniques studied thus far. (In x) 5 X dx
Evaluate the following improper integrals whenever they are convergent. 00 S 6e¹-3x dx
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
Find the present value of a stream of earnings generated over the next 2 years at the rate of 50 + 7t thousand dollars per year at time t assuming a 10% interest rate.
(a) Suppose that the graph of f (x) is above the x-axis and concave down on the interval a0 ≤ x ≤ a1. Let x1 be the midpoint of this interval, let Δx = a1 - a0, and construct the line tangent to
Evaluate the following improper integrals whenever they are convergent. S 3 +² 2 X √√x³-1 .3 dx
Evaluate the following integrals using techniques studied thus far. In x +5 dx
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
Determine the integrals in Exercises by making appropriate substitutions. et + é e fet -X et - ex dx
Approximate the value of ∫ab f (x)dx, where f (x) ≥ 0, by dividing the interval a ≤ x ≤ b into four subintervals and constructing five rectangles. (See Fig. 14.) Note that the width of the
Evaluate the following improper integrals whenever they are convergent. 00 S e -0.2x dx
Evaluate the following integrals using techniques studied thus far. sec² (x² + 1)dx x se
Determine the integrals in Exercises by making appropriate substitutions. Ji et 1 + 2ex dx
(a) Show that the area of the trapezoid in Fig. 13(a) is 1/2((h + k)∫.(b) Show that the area of the first trapezoid on the left in Fig. 13(b) is 1/2 [f(a0) + f (a1)]Δx.(c) Derive the trapezoidal
Determine the integrals in Exercises by making appropriate substitutions. ex 1 - ex dx
Evaluate the following integrals using techniques studied thus far. (2 + x²)dx 2x
Evaluate the following improper integrals whenever they are convergent. 00 1 - dx x ln x 2
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
Subdivide the interval 0 ≤ t ≤ 22 into n subintervals of length Δt = 22/n seconds. Let ti be a point in the ith subinterval.(a) Show that (R/60)Δt ≈ [number of liters of blood flowing past
Determine the integrals in Exercises by making appropriate substitutions. (1 + e¯³¹)³ 3 et dx
Evaluate the following integrals using techniques studied thus far. f(x³3/2 (x3/2 + In 2x)dx
Evaluate the following improper integrals whenever they are convergent. xp zx_ax T 00
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
In Fig. 16 a definite integral of the form ∫ab f(x)dx is approximated by the midpoint rule. Determine f (x), a, b, and n. NORMAL FLOAT AUTO REAL RADIAN CL sum(seq(1/(25-X2),
Determine the integrals in Exercises by making appropriate substitutions. multiply the numerator and denominator by e-x.] li 1 1 + ex dx
Evaluate the following integrals using techniques studied thus far. (xe ². - 2x)dx
Evaluate the following improper integrals whenever they are convergent. 00 S 2 0 x² + 1 +1 dx
In Fig. 17 a definite integral of the form ∫ab f(x)dx is approximated by the trapezoidal rule. Determine f (x), a, b, and n. NORMAL FLOAT AUTO REAL RADIAN CL (sin(22)+sum (sea(2sin
Determine the integrals in Exercises by making appropriate substitutions. multiply the numerator and denominator by e-x.] e²x 2x 1 e²x+1 2x dx
Evaluate the following integrals using techniques studied thus far. [(x²- (x² - x sin 2x)dx
Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express
Evaluate the following improper integrals whenever they are convergent. 00 √ 2x(x² + 2x(x² + 1)-3/2 dx
Figure 1 shows graphs of several functions f(x) whose slope at each x is x/√x2 + 9. Find the expression for the function f(x) whose graph passes through (4, 8). 16 8 Y 0 Figure 1 (4,8) 2 4 6 8 X
Figure 1 shows graphs of several functions f(x) whose slope at each x is x/√x + 9. Find the expression for the function f (x) whose graph passes through (0, 2). Figure 1 y (0, 2) X
Evaluate the following improper integrals whenever they are convergent. Г 00 (5x + 1)-4 dx
Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express
Figure 2 shows graphs of several functions f(x) whose slope at each x is (2√x + 1)/1x. Find the expression for the function f(x) whose graph passes through (4, 15). 15 10 5 y 0 Figure 2 2 (4, 15) 4
Figure 2 shows graphs of several functions f (x) whose slope at each x is x/ex/3. Find the expression for the function f (x) whose graph passes through (0, 6). y (0, 6) Figure 2 X
Evaluate the following improper integrals whenever they are convergent. L -00 Ax dx 4.x
Determine the following integrals using the indicated substitution. [(x + 5)-1¹/²₂ √x + 5 dx; u = √x + 5
Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express
Evaluateusing integration by parts. Ta x ex 2 dx (x + 1)²
Evaluate the following improper integrals whenever they are convergent. 0° -00 8 2 (x - 5)² xp.
Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule with n = 10. Then, find the exact value by integration and give the error for each approximation. Express
Determine the following integrals using the indicated substitution. +8 -In(x57)dx; u = ln(x5 - 7) Inv² X-5-7
EvaluateFirst, make a substitution; then, use integration by parts fxlert d dx.
Evaluate the following improper integrals whenever they are convergent. L -00 6 (1 - 3x)² dx
Consider the definite integralwhich has the value π.Suppose the midpoint rule with n = 20 is used to estimate π. Graph the second derivative of the function in the window [0, 1] by [-10, 10], and
Determine the following integrals using the indicated substitution. [x sec²x² dx x sec²x² dx; u = x²
Evaluate the following improper integrals whenever they are convergent. L -00 1 √4-x dx
In Exercises, consider the definite integralwhich has the value π.Suppose the trapezoidal rule with n = 15 is used to estimate π. Graph the second derivative of the function in the window [0, 1] by
Determine the following integrals using the indicated substitution. fa (1 + In x) sin(x ln x) dx; u = x ln x
Evaluate the following improper integrals whenever they are convergent. 00 S e -X (ex + 2)² dx
Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. .9 1 SV/ dx; n = 4
Determine the following integrals by making an appropriate substitution. [s sin x cos x dx
Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. 10 So e√x dx; n = 5
Evaluate the following improper integrals whenever they are convergent. I

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