04/19/23 Perkins

AP Calculus AB

Day 8Section 6.2

Separate variables first! 1dy

x ydx

1. Solve the differential equation:

1

1dy xdx

y

1

1dy xdx

y

1u y

du dy

ln u

1du xdx

u

21ln 1

2y x C

212ln1 x Cye e

1 y21

2 x Ce e

21

2x C

2121 xy Ce

212 1xy Ce

2. The rate of change of n with respect to t is proportional to 100 – t. Solve the differential equation.

100dn

k tdt

100dn k t dt

100dn k t dt 21

2100n t k t t C

3. passes through the point (0,10). 3

and ( )4

dyy y f t

dt

Find y.

1 3

4dy dt

y

1 3

4dy dty

34ln y t C

34ln t Cye e

34 ty Ce

34 010 Ce

10 C3410 ty e

34 t Cy e e

4

2

5

kty Ce4. Find the equation of the graph shown.

123,

4,5

312

kCe 123, 45 kCe 4,5

4

5k

Ce

312 4

5 kke

e

12

5ke

10ke

ln10k

4 ln10

5C

e

4ln10

5C

e

4

5

10C

5

10,000C

1

2,000C

ln101

2000ty e

110

2000ty

ln101

2000ty e

Perkins

AP Calculus AB

Day 8Section 6.2

1dy

x ydx

1. Solve the differential equation:

2. The rate of change of n with respect to t is proportional to 100 – t. Solve the differential equation.

3. passes through the point (0,10). 3

and ( )4

dyy y f t

dt

Find y.

kty Ce4. Find the equation of the graph shown.

123,

4

2

5

4,5