Winter wk 7 – Tues.15.Feb.05

Antiderivative = integral = area under curve

Derivative = slope of curve

Review 6.1-2: Finding antiderivatives

6.3: Introduction to differential equations

6.4: Second fundamental theorem of calculus

6.5: Equations of motion

Energy Systems, EJZ

Review Ch.6.1-2: Antiderivatives

Recall: if F=slope=df/dx =f’, then

Given a graph of the slope F=df/dx, sketch f:• Where is slope F=0? There f is flat.

• Where is slope F>0? There f increases.

• Where is slope F<0? There f decreases.

f(x)=AREA under F(x) curve.

dfF dx dx df f

dx

Analytic antiderivativesPolynomials:

Exponential and logarithmic functions:

Trigonometric functions:

1 1( )

1 ,1

n nn nd x x

n x so x dx cdx n

1,

1 1ln , ln

axax ax axde

ae so e dx e cdx a

dx so dx x c

dx x x

1cos sin , sin cos

dax a ax so ax dx ax c

dx a

6.3 Introduction to DiffEqSpeed = rate of change of position

v = ds/dtThis is a differential equation! • Do Conceptest 1, then • solve for s if v=50

Acceleration = rate of change of speeda = dv/dt

• Solve for v and s if a = -g (gravity)• See Ex.2 and 3, then • do Conceptests 2&3.

Conceptest 1

Graphs (I)-(III) show velocity versus time for three different objects.

Which travels the furthest in four seconds?

Which travels the shortest distance?

Conceptest 1 answer

Conceptest 2

Conceptest 2 answer

Conceptest 3

Conceptest 3 answer

Practice Ch.6.3 # odd-numbered probs thru #9; 10, 12, 18, 20, 22

6.4: Second Fundamental theorem

Definite integrals have limits specified, including the intercept.

Indefinite integrals have no limits specified, so the intercept is unknown.

( ) ( )b b

b

aa a

dfF dx dx f f b f a

dx

( )df

F dx dx df f x constdx

Conceptest 4

Conceptest 4 answer

Conceptest 5

Conceptest 5 answer

Conceptest 6

Conceptest 6 answer

Practice Ch.6.4 # 1, 2, 6, 8, 10, 12, 14, 18, 20

6.5 Equations of MotionYou found that if a = dv/dt = -g, then

v = -gt + C1. If the initial velocity v(t=0) = v0, then find C1:

v = -gt + _____

Displacement s = -½ gt2 + v0t + C2

If the initial position s(t=0) = s0, then find C2:

s = __________________

History and ForcesThese simple equations of motion were sought for

2000 years and finally discovered by Newton, who invented calculus for this purpose.

Forces can accelerate masses: F=maAND

masses in motion tend to stay in motion: Inertia

Ex: Recall the forces you know, and derive their potential energies.

Practice Ch.6.5 # odd-numbered probs thru #35; 36, 58