Download The Mean Value Theorem & Differential Equations

The Mean Value Theorem &
Differential Equations
3.2
Theorem 3: Rolle’s Theorem (page 237)
Suppose that y = f(x) is continuous at
every point of [a, b] and differentiable at every
point (a, b). If f(a) = f(b) = 0, then there is a
least one number c in (a, b) at which f’(c) = 0.
Theorem 4: The Mean Value Theorem
Suppose that y = f(x) is continuous on a closed
interval [a, b] and differentiable on the interval’s
interior (a, b). Then there is at least one point c in (a,
b) at which
f (b)  f (a)
f ' (c ) 
ba
Example 1: Using the Mean Value Theorem
The function f(x) = x2 is continuous for 0 ≤ x ≤ 2
and differentiable for 0 < x< 2. Find a point c in which
f (2)  f (0)
f ' (c ) 
20
Physical Interpretation
The slope of the chord of [a, b] is the average
change in f over [a, b] and f’(c) is the instantaneous
change. The Mean Value Theorem states that at some
interior point the instantaneous change must equal the
average over the entire interval.
Corollary 1: Functions with Zero Derivatives Are
Constant Functions
If f’(x) = 0 at each point of an interval I
then f(x) = C for all x in I, where C is a
constant.
Corollary 2: Functions with the Same Derivative Function on
an Interval Differ by a Constant Value There
If f’(x) = g’(x) at each point of an interval I, then
there exists a constant C such that f(x) = g(x) + C for
all x in I.
Example 2: Applying Corollary 2
Find the function f(x) whose derivative is sin x
and whose graph passes through the point (0, 2).
Differential Equations
A differential equation is an equation relating an
unknown function and one or more of its derivatives.
A function whose derivatives satisfy a differential
equation is called a solution.
For example: The function f(x) = -cos x + 3 is a
solution to the differential equation dy/dx = sin x.
Example 3: Here We Go
A heavy projectile is fired up from a platform 3
meters above the ground, with an initial velocity of 160
m/sec. Assume that the only force affecting the
projectile during its flight is from gravity which
produces a downward acceleration of 9.3 m/sec2. Find
an equation for the projectile’s height above the
ground as a function of time if t = 0 when the
projectile is fired. How high above the ground is the
projectile 3 seconds after firing?