2. To approximate the change in a function f
For a small value of delta x
The exact value of the change in f
3. To approximate f (x) at x = c (x is close to c )
4. Find the linearization for the function
f(t) = 32t – 4t2, a = 2. Use this to
approximate f(2.1).
L(t) approximates f(2.1) as 49.6. Actual value
is 49.56
5. Local Extrema a function f (x) has a:
Local Minimum at x = c if f (c) is the
minimum value of f on some open interval (in
the domain of f containing c.
Local Maximum at x = c if f (c) is the
maximum value of f on some open interval
containing c.
7. Definition of Critical Points
A number c in the domain of f is called a
critical point if either f’ (c) = 0 or f’ (c) is
undefined.
8. Find the extreme values of g(x) = sin x cos x
on [0, π]
9. Critical points: g’(x) = cos 2 x – sin 2 x
g’(x) = 0, x = π/4, 3π/4
g(π/4) = ½ , max
g(3π/4) = -1/2 , min
Endpoints (0, 0), (π, 0)
10. Assume f (x) is continuous on [a, b] and
differentiable on (a, b). If f (a) = f (b) then
there exists a number c between a and b such
that f’(c) = 0
f(c)
f(a) f(b)
a c b
11. Assume that f is continuous on [a, b] and
differentiable on (a, b). Then there exists at
least one value c in (a, b) such that
12. If f’(x) > 0, for x in (a, b) then f is increasing
on (a, b).
If f’(x) < 0 for x in (a, b), then f is decreasing
on (a, b)
If f’(x) changes from + to – at x = c, f(c) is
local maximum
If f’(x) changes from – to + at x = c, f(c) is a
local minimum
13. Concavity
f is concave up if f’(x) is increasing on (a, b)
f is concave down if f’(x) is decreasing on
(a, b)
14. If f’’(x) > 0 for x in (a, b), the f is concave up
on (a, b).
If f’’(x) < 0 for x in (a, b), the f is concave
down on (a, b).
Inflection Points – If f’’(c) = 0 and f’’(x)
changes sign at x = c then f(x) has a point of
inflection at x = c.
15. F’’(c) > 0, then f (c) is a local minimum
F’’(c) < 0, f (c) is a local maximum
F’’(c) = 0, inconclusive…may be local
max, min or neither
16. For a Function f (x):
Domain/Range of the Function
Intercepts (x and y)
Horizontal/Vertical Asymptotes
End Behavior (Polynomials)
Period, frequency, amplitude, shifts
(Trigonometric)
17. For the Derivative f’ (x):
Critical points
Increasing/Decreasing Intervals
Extrema
18. For the Second Derivative f’’ (x):
Points of Inflection
Concavity
19. Application of the Derivative involving finding
a maximum or minimum.
Example problems on page 227 (#41) and
page 228 (#43)
20. Method for finding approximations for zeros
of a function.
Uses a number of iterations to locate the
zero.
