Recursion

• Describing the present state in terms of the previous state(s)

• Modeling and solving problems involving sequential change

• Deepening understanding through another powerful representation

Example: Recursive View of Functions …

Recursive View of Functions

• Linear

• Exponential

• Polynomial

Recursive View of Functions 1

Describe patterns shown in the table.

A B

0 5

1 7

2 9

3 11

Look down the column of B’s

NEXT = NOW + 2

Bn+1 = Bn + 2

A B

0 5

1 7

2 9

3 11

Look across from A to BB = 5 + 2A

A B

0 5

1 7

2 9

3 11

Recursive View ofLinear Functions

• Explicit form: B = 5 + 2A• Recursive form:

NEXT = NOW + 2, start at 5 Bn+1 = Bn + 2, B0 = 5

• Slope seen concretely in recursive form• Rate of Change seen concretely• Note also: arithmetic sequence

Recursive View of Functions 2

Describe patterns shown in the table.

A B

0 5

1 10

2 20

3 40

Look down the column of B’s

NEXT = NOW * 2

Bn+1 = Bn * 2

A B

0 5

1 10

2 20

3 40

Look across from A to B

B = 5 * 2A

A B

0 5

1 10

2 20

3 40

Recursive View ofExponential Functions

• Explicit form: B = 5 * 2A

• Recursive form: NEXT = NOW * 2, start at 5 Bn+1 = Bn * 2, B0 = 5

• Potent comparison to linear – add constant vs. multiply by constant at each step

• Note also: geometric sequence

Common ApplicationsLinear and Exponential

Recursive point of view gives powerful information and insights

Distance-Rate-Time Compound Interest

Represent each situation with a table, graph, and equations (recursive and explicit)

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph

Money saved: $1000 initial deposit, 6% interest compounded annually

Tables and Equations

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

NEXT = NOW + 70start at 30

d = 30 + 70(t – 1)

Money saved: $1000 initial deposit, 6% interest compounded annually.

NEXT = NOW(1.06)start at 1000

Time (t) Dist (d)

1 30

2 100

3 170

4 240

A=1000×1.06t

Time (t) Amt (A)

0 1000

1 1060

2 1123.60

3 1191.02

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

Recursive Form Gives Potent

NEXT = NOW + 70

• add constant at each step

• constant difference between steps, 70 mph constant, that’s what it means to have:

constant rate of change

linear function

arithmetic sequence

repeated addition: multiplication

d = 30 + 70(t – 1)

Money saved: $1000 initial deposit, 6% interest compounded annually.

Information and Insights

NEXT = NOW(1.06)

• multiply by constant each step

• not constant difference (but constant ratio, and I wonder about quadratics), which means:

not constant rate of chg

exponential function

geometric sequence

repeated mult: exponentiationA=1000×1.06t

Dn+1 =Dn + 70 an+1 =1.06an

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

NEXT = NOW + 70

d = 30 + 70(t – 1)

Graph: line (constant rate,slope)

y-int: meaningful?For this situation, the graph starts at (1, 30)

Money saved: $1000 initial deposit, 6% interest compounded annually.

NEXT = NOW(1.06)

Graph: definitely not a line, even though it looks linear, doesn’t go up same amount for each unit over (mult, not add, a constant).

y-int: initial dep, (0, 1000)

A=1000×1.06t

Common ApplicationsLinear and ExponentialRecursive point of view gives powerful

information and insights

Distance-Rate-Time Compound Interest

Distance traveled from DT Chicago:1 hr to get out of town (30 mi), then cruise at 70 mph.

Money saved: $1000 initial deposit, 6% interest compounded annually.

Recursive View of Functions 3

Describe patterns shown in the table….

Time n Inst speed

at time nAvg spd during

each secDist fallen

during each sec

Total dist fallen after

n secs

0 sec 0 ft/sec 0 ft/sec 0 ft 0 ft

1 32 16 16 16

2 64 48 48 64

3

4

n

Skydiver falls at 32 ft/sec/sec. Ignore all other factors.

Time n

Inst speed at

time n

Avg spd during each

sec

Dist fallen during

each sec

Total dist fallen after n secs

0 sec 0 ft/sec 0 ft/sec 0 ft 0 ft

1 32 16 16 16

2 64 48 48 64

3 96 80 80 144

4 128 112 112 256

n Add 32 each sec:

32n

Start with 16, add 32 each sec:

16+32(n-1)=32n-16

32n-16 T(n)=T(n-1)+D(n)

T(n)=T(n-1)+32n-16

T(n)=(4n)2

T(n)=16n2

Quadratic Function: T(n) = an2 + bn + c

T(n) 1st difference 2nd difference

0 16 32

16 48 32

64 80 32

144 112

256

c a + b 2a

a +b + c 3a + b 2a

4a + 2b + c 5a + b 2a

9a + 2b + c 7a + b

16a + 2b+ c

Recursive View of Functions

• Linear

• Exponential

• Polynomial