209)
Determine: x(9x2+1)3dx
209)
210)
If dy
dx =x2– 4x+ 1 and y(3) = 8, find y.
210)
211)
Determine: 23q+4dq
211)
212)
Determine: 3t3– 4t2–t+ 1
t2dt
212)
213)
The present value (in dollars) of a continuous flow of income of P dollars a year for t years
at a rate of 100r percent compounded continuously is given by
t1
0
Pe–rt dt .Evaluate this
integral if P= 1000, r= 0.04, and t1= 3. Give your answer in terms of e.
213)
214)
Determine: x2+ 2x–5
x+ 3 dx
214)
215)
Determine 58x– 7
8x– 7 dx
215)
41
216)
Determine: (x2+1)2dx
216)
217)
Find the differential in terms of x and dx.
y=(4x5– 7x3 + 4)4
217)
218)
A factory is dumping garbage into a river at a rate given by dx
dt =et
200 tons per month,
where t is the time in months since the dumping began and x is the number of tons of
garbage. Find the function x(t) which gives the total number of tons of garbage dumped.
218)
219)
Determine: x
x2– 5 dx
219)
220)
Use differentials only to find an approximate value of 326.9.
220)
221)
Determine 45xdx
221)
222)
Determine:
4
1
x
2–2
xdx
222)
223)
Find the exact area of the region bounded by y=f(x) = 8 – 2x–x2 and the x–axis.
223)
224)
Suppose f(x) =
x
5
et2–et
et+tdt, then find f’(x)
224)
225)
If the marginal revenue for a manufacturer’s product is dr
dq = 700 – 6q – 8q3, find the
demand function.
225)
226)
Determine 7x6– 8x4 + 9x3
2x2dx
226)
227)
Determine: (4x3– 6x2+ 7x– 2) dx
227)
228)
A company has determined that its marginal revenue function (in dollars) is given by
C’(x) = 400 + 2x, where x is the number of units made. Find the total revenue for making 12
units by evaluating
12
0
(400 + 2x) dx .
228)
229)
The demand equation for a certain product is p= 200 –q2, where p is the price per unit (in
dollars) for q units. If its supply equation is p= 6q+ 160, find the consumers’ surplus and
the producers‘ surplus when market equilibrium has been established.
229)
230)
A bacteria population is increasing at a rate of dp
dt = 3t22t3. Find p as a function of t.
230)
43
231)
Determine: (5x4+ 4x2– 6x+ 5) dx
231)
232)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
232)
233)
Evaluate:
11
5
e ln x
xdx
233)
234)
If y” =ex– 2, and y'(0) =3 and y(0) = 4, find y.
234)
235)
Determine: 1
4x
dx
235)
236)
The rate of change in cost of producing a product is given by C’(x) =x2
x3+ 5
+ 2x. Find the
increase in cost from producing 5 units to producing 15 units.
236)
44
237)
Determine: x2(2x3–5)4dx
237)
238)
The income (in dollars) from a fast food chain is increasing at a rate of f(t) = 10,000e0.02t
where t is in years. Find
3
0
10,000e0.02tdt, the total income for the chain over the first
three years.
238)
Explanation:
239)
The base of a triangle is decreasing at a rate of db
dt = – 3e–t
t. Find b as a function of t.
239)
Explanation:
240)
Determine y2(y2/3 + 3y) dy
240)
Explanation:
241)
Determine: –1
2 – 3xdx
241)
Explanation:
45
Explanation:
242)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
242)
243)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
243)
244)
Determine: e7–2x dx
244)
245)
Determine (5x2+ 4x)e5x3+6x2+7dx
245)
46
246)
Suppose the surface area of a 1.7 meter–tall person changes at the rate of ds
dx = 0.13775
x–0.58 square meters per kilogram, where x is the weight in kilograms. Find the increase in
surface area for a person whose weight increases from 70 kilograms to 75 kilograms by
evaluating
75
70
0.13775x–0.58 dx.
246)
247)
The marginal revenue for a product is given by dr
dx =
–30
(2x+1)2+ 30. Find r.
247)
248)
The supply equation for a company is q= 4p3– 2p. Find dp
dq from dq
dp .
248)
249)
Determine: x2+ 6x– 3
xdx
249)
250)
Find the exact area of the region bounded by the graphs of y= 9 –x2 and y= 5 – 3x. Also
sketch the region.
250)
47
251)
The marginal price for a weekly demand of x bottles of shampoo is given by p’(x) =
–3
(3x +50)2. Find p(x).
251)
252)
Find y subject to the given conditions: y” =1
3x(2/3) + 5; y’(0) = 1; y(0) = 5
252)
253)
Determine: x2
4x3– 3 dx
253)
254)
Determine: (x3–x+ 1)(x4– 2x2+ 4x)–5dx
254)
255)
The length of a side of a square is increasing at a rate of ds
dt =4
3t1/3(t2/3 + 9) . Find s as a
function of t.
255)
256)
The marginal revenue function for a manufacturer’s product is given by dr
dq =450
(q+5)2.
Determine the demand equation for the product.
256)
257)
Suppose
6
1
f(x) dx = 9;
4
6
f(x) dx = 4, then find
4
1
f(x) dx .
257)
258)
Suppose that in the United States the rate of change of the percentage of workers who
belong to unions can be modeled by dy
dx = 0.15(0.2x+3)2– 1.4(0.2x+ 3), where x is the
number of years after 1940. Find y.
258)
259)
Evaluate:
2
1
d
dx
5
3
e x +x3 dx dx
259)
260)
If y=x, use differentials to approximate the change in y if x changes from 25 to 25.3.
260)
261)
Find the differential in terms of x and dx.
y= ln (6x3+ 5x2)3
261)
262)
Suppose the rate of savings in a country is given by dS
dt = 2.7t2– 400t + 9000, where t is the
time in years, and S is the amount of money saved in billions of dollars. Find the function
S(t) which gives the total amount of money saved. You may assume that $0 is saved at t=
0.
262)
263)
Find the exact area of the region bounded by y=x2+ 1, x= 1, x= 3, and the x–axis. Also
sketch the region.
263)
264)
The supply equation for a company is q=300 +p. Find dp
dq from dq
dp .
264)
265)
Find the area of the region bounded by the given equations:
y=x2 + 6x + 10 and y=x + 6
265)
266)
Suppose the rate of savings in a country is given by dS
dt = 3.3t2– 900 t + 32,891.60, where t
is the time in years, and S is the amount of money saved in billions of dollars. Find the
function S(t) which gives the total amount of money saved. You may assume that $0 is
saved at t= 0.
266)
267)
An oil company estimates that for the next few months, oil will be pumped (in thousands
of gallons per month) from a producing field at the rate of R(t) =2t
(t2+1)3+ 5, where t is in
months. Find 2t
(t2+1)3+ 5 dt.
267)
50
268)
Find the area between the graph of y=f(x) =x3+x+ 3 and the x–axis on the interval
2,5 . Use your graphing calculator to see a graph of this area.
268)
269)
Use the trapezoidal rule with n= 4 to estimate the value of
2
0
x2dx
269)
270)
Suppose the rate of change in the number of violent crimes per 100,000 people in the
United States can be modeled by dc
dt =280
2t+ 6 . Find c(t).
270)
271)
Determine 4x2 + 4x + 2
2x – 1 dx
271)
272)
A manufacturer’s marginal cost function is dc
dq = 0.6(0.2q–20)2, where c is the total cost (in
dollars) of producing q units of a product. If the manufacturer increases output from 50
units to 100 units, determine the change in total cost.
272)
273)
Determine: e2x+e–2x+e2dx
273)
274)
A yeast culture is growing at a rate of A’(t) = 0.3e0.2t2, where t is the time in hours and A(t)
is the amount in grams. Use n= 8 and the trapezoidal rule to approximate
4
0
0.3e0.2t2dt,
the amount the culture grew over the first four hours.
274)
275)
A manufacturer’s marginal cost function is dc
dq = .1q+ 7. Determine the cost involved to
increase production from 40 to 60 units.
275)
276)
The supply equation for a certain radio is given by p= 0.8 x+ 17 where p is the price in
dollars and x is the number of radios supplied. Use differentials to approximate the price
when 620 radios are supplied. (Hint: Use x = 625.)
276)
277)
Use Simpson’s rule with n= 4 to estimate the value of
1
0
4xdx
277)
278)
Evaluate:
2
0
f(x) dx where f(x) =2x+ 1 if x 1
3xif x 1
278)
279)
Determine:
4
1
x x dx
279)
280)
Determine: (3x–4)5dx
280)
281)
Determine:
2
2
3x
x2– 1 dx
281)
52
282)
Suppose that the points (1, 1), (2, 4), and (3, 2) lie on the graph of the continuous function f,
where f(x) 0. Use the trapezoidal rule and all of the given points to approximate the area
between the graph of f and the x–axis on the interval 1, 3 .
282)
283)
A bacteria population is increasing at a rate of dp
dt = 8t7t2. Find p as a function of t.
283)
284)
Scientists use the formula S=kW2/3 to determine an animal’s surface area S (in square
meters) from its weight W (in kilograms), where k is a constant that varies from animal to
animal. Suppose the scientist studies a certain animal with k= 0.1. Use differentials to
approximate the surface area if the animal weighs 124 kilograms. (Hint: Use W= 125.)
284)
285)
It is estimated that service industries will grow at the rate of R’(t) = 5e1/(t+1) where t is the
time in decades and R(t) is the percent of (non–farming) workers who are employed in the
service industry. If 30 percent of the (non–farming) workers are now in the service
industry, then use the trapezoidal rule and
n= 4 to approximate 30 +
2
0
5e1/(t+1) dt, the percentage of workers who will be working
in the service industry in 2 decades.
285)
286)
Use differentials to approximate: e0.2
286)
287)
Use Simpson’s rule with n= 4 to estimate the value of
2
0
2xdx
287)
288)
Determine: p
5–1
2dp
288)
53
289)
Use differentials to approximate the change in the wattage W of a flood light with a
resistance R= 8 ohms, if the current I is increased from 4 amperes to 4.2 amperes. (Hint:
Use W=RI2.)
289)
290)
Use differentials only to find an approximate value of 3.9.
290)
291)
Use your graphing calculator to find the area between the x–axis and f(x) = – x2+ 4x– 3 on
the interval 1 x 3. Verify your result by finding
3
1
(–x2+ 4x– 3) dx.
291)
292)
Find the area between the graph of y=f(x) =x3, and the x–axis on the interval 0,2 . Use
your graphing calculator to see a graph of this area.
292)
293)
Find the area of the region bounded by the curves y=x2 and y=x+ 2 from x= 1 to x= 2.
293)
294)
The acceleration of an object after t seconds is given by y” = 5et+ 6 t, the velocity at 4
seconds is given by y’(4) = 5e4+ 33 feet/sec, and the position at 0 seconds is given by
y(0) = 0 feet. Find y(t).
294)
295)
Determine: 8(1 – 4x) dx
295)
296)
Determine (x2+ 4x)ln(x3+ 6x2+ 1)
x3+ 6x2+ 1 dx
296)
54
297)
Determine: 4
9 + 7xdx
297)
298)
The producers of a long running television show expect the number of viewers (in
thousands) to decrease at a rate of V‘(x) = – 20 – 3x, where x is the number of weeks after
the television season begins. Find the total decrease in viewers in the first three weeks by
evaluating
3
0
(–20 – 3x) dx .
298)
Explanation:
299)
After weeks of television promotion, an electronics store finds that its sales change at a rate
given by S’(t) = – t8t2+ 9 + 20t+ 20, where t is the number of days after the advertising
ends and S is in thousands of dollars. Find the total sales for the first week after the
advertising ends. Find
7
0
(–t8t2+ 9 + 20t + 20 ) dt.
299)
Explanation:
300)
Use differentials to approximate 38.1.
300)
Explanation:
301)
Determine: e4x+3dx
301)
Explanation:
302)
Determine:
1/4
0
(1 – 4x)4dx
302)
Explanation:
303)
If y’ = 6x2– 4x– 3 and y(1) = 2, find y.
303)
Explanation:
Explanation: