# Calculus (Lec02)

## yukita@k.hosei.ac.jp

### •Derivative

### •Indefinite Integral

### •Definite Integral

### •Taylor Expansion

### •Differential Equation

### •Applications

#### •Tangent

Let f(x) be a cubic function given below.

Let y=g[x,a] be the tangent line at point (a,f(a)). g[x,a] is given as a linear function of x as follows.

Let us have a=1 and plot the curve and its tangent simultaneously.

We see that point (1,f(1) ) is a double intersection point of the curve y=f(x) and its tangent at (1,f(1)). Considering the degree of equations, we can expect the existence of another intersection point of the curve and the tangent.

#### •Area

Let us compute the area of the domain surrounded by the curve y=f(x) and its tangent at (1,f(1)).

### •Exercises

#### •Problem 1

Let f(x)=. Find the tangent at (1,f(1)).

##### •Solution

#### •Problem 2

Find all the intersections of the curve and the tangent in problem 1.

##### •Solution

Converted by *Mathematica*
(April 21, 2003)