Can somebody help me with these two calculus questions? (100 points. Trash responses = report)

(a) Assume that f(x) is a real-valued function defined for all real numbers x

on an open interval whose center is a certain real number a. What does

it mean to say that f(x) has a derivative f

0

(a) at x = a, and what is the

value of f

0

(a)? (Give the definition of f

0

(a).)

(b) Use the definition of f

0

(a) you have just given in part (a) to show that if

f(x) = 1

2x − 1

then f

0

(3) = −0.08.

(c) Find lim

h→0

sin7

π

6 +

h

2

−

1

2

7

h

.

2. Let I be a bounded function on R and define f by f(x) = x

2

I(x). Show that

f is differentiable at x = 0.

3. Use the definition of derivative to find f

0

(2) for f(x) = x +

1

x

.

4. If g is continuous (but not differentiable) at x = 0, g(0) = 8, and f(x) = xg(x),

find f

0

(0).

5. (a) State the definition of the derivative of f(x) at x = a.

(b) Using the definition of the derivative of f(x) at x = 4, find the value of

f

0

(4) if f(x) = √

5 − x.

6. Let f be a function that is continuous everywhere and let

F(x) = f(x) sin2 x

x

if x 6= 0,

0 if x =

the answer is; biodiversity resulting from few ancestors

adaptive radiation occurs when a population is subjected to a large natural environment with spatial ecological variations. the parts of the population in the different ecologies begin to adapt to their local environments. due to non-random mating across the large populations, this causes the different populations from the large population to speciate within their local environments. they, therefore, diverge to different species sharing a common ancestor.