209)
Suppose the rate of savings in a country is given by dS
dt = 5.1et + 400t2 + 6000, 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.
209)
210)
Determine 2x2 + 13x + 13
x + 5 dx
210)
211)
Find the exact area of the region bounded by y=f(x) =ex + x, x= 0, x= 2, and the x–axis.
211)
212)
Determine: x2– 4x+ 7
x+ 1 dx
212)
213)
A company has determined that its marginal cost function (in dollars) is given by
C’(x) = 400 + 2x, where x is the number of units made. Find the total cost for making 12
units by finding the area in the first quadrant bounded by y=C’(x) = 400 + 2xand the lines
y= 0, x= 0, and x= 12.
213)
214)
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).
214)
215)
215)
41
217)
A bacteria population is increasing at a rate of dp
dt = 3t22t3. Find p as a function of t.
217)
218)
The marginal revenue for a product is given by dr
dx =
(2x+1)2+ 30. Find r.
218)
219)
Use differentials only to find an approximate value of 326.9.
219)
220)
Find the area of the region bounded by the curves y=x2 and y=x+ 2 from x= 1 to x= 2.
220)
221)
The velocity of an object is given by V(t) =12
t t2– 1
, where t is the time in seconds and V(t)
is the speed in feet/sec. Use n= 6 and Simpson’s rule to approximate
5
2
12
t t2– 1
dt, the
distance moved over the interval 2, 5 .
221)
222)
If y’ = 6x2– 4x– 3 and y(1) = 2, find y.
222)
223)
Find y subject to the given conditions: y” =1
3x(2/3) + 5; y‘(0) = 1; y(0) = 5
223)
216)
216)
224)
Evaluate the definite integral
2
0
2xdx by first finding out Sn and then taking limit as n
. (Use the right–hand endpoint of each sub–interval.)
224)
225)
Determine: 4
9 + 7xdx
225)
226)
The velocity of an object is given by V(t) =12
t t2– 1
, where t is the time in seconds and V(t)
is the speed in feet/sec. Use n= 6 and the trapezoidal rule to approximate
5
2
12
t t2– 1
dt,
the distance moved over the interval 2, 5 .
226)
227)
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.
227)
228)
Find the indefinite integral x2e+exdx.
228)
229)
Determine: x2(x3+ 2x2– 5) dx
229)
43
230)
Evaluate:
11
5
e ln x
xdx
230)
231)
The acceleration of an object after t seconds is given by y” = 32, the velocity at 0 seconds is
given by y’(0) = 0 feet/sec, and the position at 2 seconds is given by y(2) = 64 feet. Find y(t).
231)
232)
Determine: 2x+ 3
4dx
232)
233)
Determine: z2– 3z+ 5
z+ 2 dz
233)
234)
An oil tanker is losing oil at a rate of R’(t) =60
t2+ 9
where t is the time in minutes and R(t)
is the radius of the oil slick in feet. Use n= 6 and Simpson’s rule to approximate
5
0
60
t2+ 9
dt,the size of the radius after 5 seconds.
234)
235)
Determine: (2x+ 1)e(2x+1)2dx
235)
236)
The producers of a long running radio show expect the number of listeners (in hundreds)
to decrease at a rate of L’(x) = – 5 –x2, where x is the number of weeks after the first day of
the year. Find the total decrease in listeners in the first 2 weeks by evaluating
2
0
(–5 –x2) dx .
236)
237)
Determine: (2x+ 3)ex2+3x+5dx
237)
238)
Use differentials to approximate e0.02.
238)
239)
The supply equation for a certain radio is given by p= 0.5 x+ 12 where p is the price in
dollars and x is the number of radios supplied. Use differentials to approximate the price
when 1604 radios are supplied. (Hint: Use x = 1600.)
239)
240)
Determine: x(9x2+1)3dx
240)
241)
If y’ =x+ 2 and y(2) = 5, find y.
241)
242)
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.
242)
243)
Find the differential in terms of x and dx.
y= ln (6x3+ 5x2)3
243)
244)
Find the differential in terms of x and dx.
y= 3x2– 5x + 4
244)
245)
If the rate of change of a company’s revenue can be modeled by dr
dt = 3t2, then find r(t) the
revenue function.
245)
246)
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.
246)
247)
Find the exact area of the region bounded by y=x2+ 1, x= 1, x= 3, and the x–axis. Also
sketch the region.
247)
248)
Find dy if y=2
x+ 3 .
248)
249)
The radius of a circle is growing at a rate of dr
dt =5
(t+ 1) ln(t+ 1) . Find r as a function of t.
249)
250)
For a group of rats that were fed a particular diet, the rate of change of the average weight
gain G (in grams) of a rat with respect to the percent P of yeast in the diet is
dG
dP = – P
32 + 2; 0 P 100. If G= 36 when P= 8, find G.
250)
251)
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 .
251)
252)
Determine: 4x9 – 3x2dx
252)
253)
The acceleration of an object after t seconds is given by y” = 5t– 6, the velocity at 4 seconds
is given by y’(4) = 18 feet/sec, and the position at 1 second is given by y(1) = 0 feet. Find
y(t).
253)
254)
The producers of a new television show expect the number of viewers (in thousands) to
increase at a rate of V‘(x) = 100 + 10x, where x is the number of weeks after the show
premiers. Find the total increase in viewers in the first seven weeks by evaluating
7
0
(100 + 10x) dx .
254)
255)
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.
255)
47
256)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
256)
257)
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.
257)
258)
Evaluate:
2
0
f(x) dx where f(x) =2x+ 1 if x 1
3xif x 1
258)
259)
After weeks of radio promotion, a shoe store finds that for the next 30 days its sales change
at a rate given by S‘(t) = – 12t2+ 300t, where t is the number of days after the advertising
ends and S is in dollars. Find the total sales for the second week after the promotion ends.
Find
14
7
(–12t2+ 300t) dt.
259)
260)
Find the area of the region bounded by the given equations:
y=x2 + 6x + 10 and y=x + 6
260)
48
261)
For the region bounded by f(x) = 4 –x, y= 0, x= 0, and x= 3, approximate the area by
evaluating S6. (Use the right–hand endpoint of each subinterval.)
261)
262)
Evaluate:
1
0
x2+ 3x+ 2
x2+ 2x+ 1 dx
262)
263)
The height of a cylinder is decreasing at a rate of dh
dt = – 12t2et3. Find h as a function of t.
263)
264)
Determine:
8
4
8 –xdx
264)
265)
If the rate of change of a company’s revenue can be modeled by dr
dt = 712t, then find r(t) the
revenue function.
265)
266)
If y= ( x2+ 3x)4, find dx
dy .
266)
267)
Determine 5x4– 7x2 + 9
x2dx
267)
268)
Use differentials to approximate: 25.6.
268)
269)
Determine: (x3+3x) dx
269)
270)
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.
270)
271)
Determine x2– 8x + 6
x– 4 dx
271)
272)
Determine: x2
4x3– 3 dx
272)
273)
After weeks of radio promotion, a shoe store finds that its sales change at a rate given by
S’(t) = – 12t2+ 300t, where t is the number of days after the advertising ends and S is in
dollars. Find the total sales for the first week after the promotion ends.
Find
7
0
(–12t2+ 300t) dt.
273)
274)
If q=p3– 6p2+ 3p– 8, find dp
dq when p= 1.
274)
275)
Determine –2x·79–x2dx
275)
50
276)
A bacteria population is increasing at a rate of dp
dt = 4t39t4. Find p as a function of t.
276)
277)
Determine: x2+ 2x–5
x+ 3 dx
277)
278)
Suppose
6
1
f(x) dx = 9;
4
6
f(x) dx = 4, then find
4
1
f(x) dx .
278)
279)
The demand equation for a product is p=49 – 6q and the supply equation is p= 1 +q.
Determine the consumers’ surplus and the producers’ surplus.
279)
280)
The demand equation for a certain product is p= 30 – 0.1q, where p is the price per unit (in
dollars) for q units. If its supply equation is p= 0.15q+ 5, find the consumers’ surplus and
the producers‘ surplus when market equilibrium has been established.
280)
281)
Find the indefinite integral 5x–3
5x
dx.
281)
282)
Determine e
–2
x2
x3dx
282)
51
283)
Use differentials to approximate: ln(0.95).
283)
284)
Determine:
4
–4
3zez2+ 3 dz
284)
285)
Find the area between the graph of y=f(x) =e0.3x, and the x–axis on the interval 0,5 . Use
your graphing calculator to see a graph of this area.
285)
286)
Use Simpson’s rule with n= 4 to estimate the value of
1
0
4xdx
286)
287)
Evaluate:
11
5
e3dx
287)
288)
Determine: 2(x2+ 5)
3dx
288)
289)
Use differentials and C= 250 + 0.30x to approximate the change in the cost C (in dollars) to
produce x pounds of candy if the number of pounds of candy increases from 10 pounds to
10.6 pounds.
289)
290)
Use Simpson’s rule with n= 4 to estimate the value of
2
0
2xdx
290)
291)
Find y subject to y’ = (1 + 7x)2; y(0) = 1.
291)
292)
Express the area of the shaded region in terms of an integral (or integrals). Do not evaluate
the expression:
292)
Explanation:
293)
If dy
dx =x2– 4x+ 1 and y(3) = 8, find y.
293)
Explanation:
294)
Evaluate:
2
1
2xdx
2
–
2
1
(2x)2dx
294)
Explanation:
295)
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.
295)
Explanation:
53
Explanation:
296)
The rate of change of the value of a house that cost $250,000 to build can be modeled by dV
dt
= 7.5e0.075t, where t is the time in years since the house was built, and V is the value (in
hundreds of thousands of dollars) of the house. Find V(t).
296)
297)
Find y subject to the given conditions: y” = 6x– 2; y’(1) = 2; y(1) = 2.
297)
298)
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 a rate of R(t) =3t2
(t3+1)4+ 9, where t is in
months. Find 3t2
(t3+1)4+ 9 dt.
298)
299)
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.
299)
300)
Use differentials only to find an approximate value of 4.1.
300)
301)
Determine: x
4x2+ 1
dx
301)
302)
Determine: ln(xe2x)
xdx
302)
303)
The producers of a new radio show expect the number of listeners (in hundreds) to
increase at a rate of L’(x) = 17 +x2, where x is the number of weeks after the show
premiers. Find the total increase in listeners in the first four weeks by evaluating
4
0
(17 +x2) dx .
303)
304)
Find the exact area of the region bounded by y=f(x) =2x + 5, x= – 2, x= 2, and the
x–axis.
304)
305)
Suppose that the points (3, 2), (3.25, 2.5), (3.5, 2.25), (3.75, 2)and (4, 1.5) lie on the graph of
the continuous function f, where f(x) 0. Use Simpson’s rule and all of the given points to
approximate the area between the graph of f and the x–axis on the interval 3, 4 .
305)
306)
If the marginal revenue for a manufacturer’s product is dr
dq = 700 – 6q – 8q3, find the
demand function.
306)
307)
Use differentials to approximate: e0.2
307)
308)
Use the trapezoidal rule with n= 3 to estimate the value of
3
0
1
x2+ 1 dx
308)
309)
Use definite integrals to find the area between y = – x2, the x–axis, x= – 1, and x= 1.
309)