You will need

- mathematical Handbook;
- - notebook;
- - pencil;
- - handle;
- - protractor;
- a pair of compasses.

Instruction

1

Please note that the graph of a differentiable function f(x) at point x0 has no differences to the segment of the tangent. So it is rather close to the segment l passing through the points (x0, f(x0)) and (x0+Δx f(x0 + Δx)). To specify the straight line passing through the point A with the coefficients (x0, f(x0)), specify its angular coefficient. In this case it is equal to Δy/Δx secant tangent (DF→0) and tends to f‘(x0).

2

If the values of f‘(x0) does not exist, it is possible that a tangent there, or it is held vertically. On this basis, the presence of the derivative of the function at the point x0 due to the existence applied on other than vertical tangent, which touches the graph of the function at the point (x0, f(x0)). In this case, the slope of the tangent is f'(x0). Becomes clear the geometric meaning of derivative, that is, the calculation of the angular coefficient of the tangent.

3

That is, in order to find the slope of the tangent, you need to find the value of the derivative of the function at the point of tangency. Example: to find the slope of the tangent to the graph of the function y = X3 at the point with abscissa X0 = 1. Solution: Find the derivative of this function y(x) = 3x2; find the value of the derivative at the point X0 = 1. have(1) = 3 × 12 = 3. The slope of the tangent at the point X0 = 1 is equal to 3.

4

Draw the picture for more tangents so that they adjoined to the graph of the function at the following points: x1, x2 and X3. Mark the angles that are formed by tangent data with the abscissa axis (angle measured in the positive direction from the axis to the tangent line). For example, the first angle α1 is an acute, the second (α2) is stupid, but the third (α3) will be zero, as performed the straight line is parallel to the OX axis. In this case, the tangent of the obtuse angle has a negative value, and tangent of an acute angle is positive, when tg0 and the result is zero.

# Advice 2: How to carry out the tangents to circles

A tangent to a given circle is called a straight line that has only one common point with the circle. Tangent to a circle is always perpendicular to its radius drawn to the tangent point. If two tangents drawn from one point, not belonging to a circle, then the distance between this point and the touch points will always be the same. Tangents to

**circles**are constructed in different ways, depending on their location relative to each other.Instruction

1

The construction of the tangent to the same circle.

1. Built a circle of radius R and take a point A, to go through the tangent.

2. Construct the circle with center in the midpoint of the segment OA and the radius equal to half of this segment.

3. The intersection of two circles are the points of tangency of tangents drawn through point A to a given circle.

1. Built a circle of radius R and take a point A, to go through the tangent.

2. Construct the circle with center in the midpoint of the segment OA and the radius equal to half of this segment.

3. The intersection of two circles are the points of tangency of tangents drawn through point A to a given circle.

2

External tangent to two

1. Built two circles of radius R and r.

2. Is a circle of radius R – r centered at the point O.

3. To the resulting circle is tangent from point O1, the point of tangency is marked with the letter M.

4. The radius R passing through the point M indicates the point T – the point of tangency of a great circle.

5. Through the center O1 of the small circle is of radius r parallel to the radius R of the large circle. The radius r indicates the point T1 to the point of tangency of the small circle.

6. Video TT1 – tangent to the given

**circles**.1. Built two circles of radius R and r.

2. Is a circle of radius R – r centered at the point O.

3. To the resulting circle is tangent from point O1, the point of tangency is marked with the letter M.

4. The radius R passing through the point M indicates the point T – the point of tangency of a great circle.

5. Through the center O1 of the small circle is of radius r parallel to the radius R of the large circle. The radius r indicates the point T1 to the point of tangency of the small circle.

6. Video TT1 – tangent to the given

**circles**.3

Internal tangent to two

1. Built two circles of radius R and r.

2. Is a circle of radius R + r centered at the point O.

3. To the resulting circle is tangent from point O1, the point of tangency is marked with the letter M.

4. OM the ray intersects the first circle at a point T in the tangent point of a great circle.

5. Through the center O1 of the small circle is of radius r parallel to the ray OM. The radius r indicates the point T1 to the point of tangency of the small circle.

6. Video TT1 – tangent to the given

**circles**.1. Built two circles of radius R and r.

2. Is a circle of radius R + r centered at the point O.

3. To the resulting circle is tangent from point O1, the point of tangency is marked with the letter M.

4. OM the ray intersects the first circle at a point T in the tangent point of a great circle.

5. Through the center O1 of the small circle is of radius r parallel to the ray OM. The radius r indicates the point T1 to the point of tangency of the small circle.

6. Video TT1 – tangent to the given

**circles**.