MATH 1210 - Section 002

Fall 2010

Nicolas Fourrier

Due on November 18, 2010

Homework 9 Name:

1. Find the indefinite integral: R (a) 8x9 − 3x6 + 12x3 dx R (b) x(2 − x)2 R √ √ (c) 5 6 − 6 x RRR x (d) e dx R x5 −x3 +2x (e) x4

(a)

R

8x9 − 3x6 + 12x3 dx = 54 x10 − 37 x7 + 3x4 + C

x(2 − x)2 = 2x2 − 34 x3 + 14 x4 + C R √ √ (c) 5 6 − 6 x = 4x3/2 − 67 x7/6 + C RRR x (d) e dx = ex + 12 Ct2 + Dt + E R x5 −x3 +2x (e) = 12 x2 − ln |x| − x12 + C x4 (b)

R

2. Solve the initial value problems: (a) f 0 (x) = 2x + 1; f (1) = 6 (b) f 00 (x) = 2 − 12x; f (0) = 9; f (2) = 15 (c) f 00 (x) = 24x2 + 2x + 10; f 0 (1) = −3; f (1) = 5 √ (d) f 0 (x) = x(6 + 5x); f (1) = 10 (e) f 0 (x) =

1 (x2 x

− 1); f (1) =

1 2

(a) f (x) = x2 + x + 4 (b) f (x) = x2 − 2x3 + Cx + D ⇒ f (x) = x2 − 2x3 + 9x + 9 (c) f (x) = 2x4 + 31 x3 + 5x2 − 22x +

59 3

(d) f (x) = 4x3/2 + 2x5/2 + 4 (e) f (x) = 12 x2 − ln x if x > 0 and f (x) = 12 x2 − ln −x −

1 2

if x < 0

3. A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92 − 0.002x. If the cos of producing one item is $562, find the cost of producing 100 items. Marginal cost = 1.92 − 0.002x = C 0 (x), so C(x) = 1.92x − 0.001x2 + K. But C(1) = 562, thus K = 560.081. Therefore, C(x) = 1.92x − 0.001x2 + 560.08 and C(100) = 742.081

4. A car is traveling at 100km/h when the driver sees an accident 80 meters ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup? The acceleration/deccelartion is the second derivative of the position (the first derivative being the velocity) Let the acceleration be a(t) = k km/h2 . We have v(0) = 100 km/h and we can take the initial position s(0) = 0. We want the time tf for which v(t) = 0 to satisfy s(t) < 0.08km. In general, v 0 (t) = a(t) = k, so v(t) = kt + C, where C = v(0) = 100. Now, s0 (t) = v(t) = kt + 100, so s(t) = 12 kt2 + 100t + D, where D = s(0) = 0. Thus, s(t) = 21 kt2 + 100t. Since v(tf ) = 0, we have ktf + 100 = 0. Therefore, s(tf ) = 12 k(− 100 )2 + 100(− 100 ) = −5000 , but s(t) < 0.08 implies that k < − 5000 . So the constant k k k 0.08 of decceleration k must be less than −62500 km/h2 . 5. The llinear denity of a rod of lengty 1 meter is given by p(x) = √1x , in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod. Let the mass, measured from one end, be m(x), then m(0) = 0 and p = dm = x−1/2 , thus m(x) = 2x1/2 + C dx √ and m(0) = C = 0. Therefore m(x) = 2 x, so the mass of the 100-centimeter rod is 20g.

6. Find the indefinite integral, using the substitution rule. Credit will be given for the details R √ (a) ex 1 + ex dx R (b) x3 (2 + x4 )5 dx R dx (c) (1−6t)4 √ R (d) (x + 1) 2x + x2 dx R x (e) dx (x2 +1)2 R (ln x)2 dx (f) x

(a)

R

√ ex 1 + ex dx = 23 (1 + ex )3/2 + C by letting u = 1 + ex

(b)

R

x3 (2 + x4 )5 dx =

(c)

R

(d)

R

1 = 18(1−6x) 3 + C by letting u = 1 − 6x √ 2 (x + 1) 2x + x dx = 13 (2x + x2 )3/2 + C by letting u = 2x + x2

(e)

R

x dx (x2 +1)2

(f)

R

(ln x)2 dx x

1 (2 24

+ x4 )6 + C by letting u = 2 + x4

dx (1−6x)4

=

−1 2(x2 +1)

+ C by letting u = x2 + 1

= 31 (ln x)3 + C by letting u = ln x