11.Taylor’s formula.

Answer:

A polynomial is of the form P(x)=Q₀+Q₁(x-x₀)+…+a_n(x-x₀)ⁿ (1)

where Q₀,Q₁,…Q_n and x₀ are constants.

In particular, a constant polynomial P(x)=Q₀ is of degree zero, if Q₀ 0

If f is differentiable all x₀ , then f(x)=f(x₀)+f’(x₀)(x-x₀)+ (x-x₀)

Where =0

The polynomial T₁(x)=f(x₀)+f’(x₀)(x-x₀) which is of degree 1 and satisfies T₁(x)=f(x₀) T^’₁(x₀)=f’(x₀), approximates f so well near x₀ that

=0 (2)

Now suppose that f has n derivatives at x₀ and T_n is the polynomial of degree n such that

T_n^(r)(x₀)=f^(r)(x₀) 0 r n (3)

T_n is polynomial of degree n;

T_n(x)=a₀+a₁(x-x₀)+…+a_n(x-x₀)ⁿ (4)

Differentiating (4) yields T_n^(r)(x₀)=r!a_r

So (3) determines a_r uniquely as a_r=f^(r)(x₀)/r!

Therefore,

T_n(x)=f(x₀)+ (x-x₀) + (x-x₀)² + …+ (x-x₀)ⁿ = (x-x₀)^r

We callT_n the n-th Taylor polynomial of f about x₀ (xxx дегенимиз x³)

1. sinx=x – +– -– +….

2. cosx=1 – – + – – – + – - …

3. ln(1+x)=x – x²/2 + x³/3 – x⁴/4 + x⁵/5 - …

4. e^x= 1+ x + x²/2! +x³/3! + x⁴/4! + …

5. (1+x)^a=1 + ax + a(a-1)x²/2! + a(a-1)(a-2)x³/3! + … + x^a

12.Analysis of function using the derivative plotting graph of function.

Answer:

To construct the graph of the function y=f(x) to find:

1. Domain and Range

2. Symmetries f is said to be an even function (if f(-x)=f(x)) , and is said to be an odd function (if f(-x)=-f(x))

3. Points of discontinuity. The line x=a is called a vertical asymptote of the curve y=f(x), if =

4. Asymptotes (oblique or slant)

Y=kx + b, k= , b= if k=0, then y=b is a horizontal asymptote.

5. X-intercepts (zeros of function and the region of constant sign)

6. Relative extrema (max and min).

If f’(x)>0, for every value ox x in (a;b) , then f is increasing on [a;b].

If f’(x)<0, for every value ox x in (a;b), then f is decreasing on [a;b].

If f has a relative extremum at x=x₀ is a critical point of f; that is , either f’(x₀)=0 or f is not differentiable at x₀.

7. Concavity , inflection points.

1. If f’’(x)>0 , for every value of x, then f is concave up on that interval.

2. If f’’(x)<0 , for every x in the open interval , then f is concave down on that interval.

If a function f changes the direction of its concavity at the point

(x₀ ,f(x₀)) , then we say that f has an inflection point at x₀

13. Integration. Indefinite integral.

Answer:

Definition. A function f is called an Antiderivative of a function f on the given open interval, if F’(x)=f(x) for all x in the interval.

For example, the function F(x)= x⁴ is an antiderivative of f(x)=x³ on the interval (- ;+ ), because for each z in this interval. F’(x)=( x⁴)’=x³=f(x).

However, F(x)= x⁴ is not the only antiderivativeof f. If we add any constant C to x⁴ , then the function x⁴ + C is also an antiderivativeof f on (- ;+ ).

Since, ( x⁴ + C) = x³ = f(x).

If F(x) is any antiderivative of f(x), then F(x) + C is also an antiderivative on that interval. The process of finding antiderivative is called integration.

= F(x) + C.

³ dx = x⁴/4 + C.

The expression is called an indefinite integral.

Equation (1) should be real as:

“The integral of f(x) with respect to x is equal to F(x) plus a constant. ”

Integration formulas:

1. =x + C; 9) =arcsin + C;

2. ⁿdx= xⁿ⁺¹ /n+1 + C; 10) a² +x²= arctg + C;

3. =sinx + C; 11) =ln|x + | + C;

4. =-cosx + C; 12) a² -x²= ln| + C;

5. cos²x=tgx + C; 13) = ln|x| + C;

6. sin²x=-ctgx + C; 14) = + a²/2arcsin + C;

7. ^xdx = e^x + C; 15) = + a²/2ln|x + | + C.

8. ^xdx = a^x/lna + C;