Calculus (Lec02)

yukita@k.hosei.ac.jp

•Derivative

f[x_] = Sin[x]

Sin[x]

Plot[f[x], {x, -Pi, Pi}]

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D[f[x], x]

Cos[x]

f '[x]

Cos[x]

Plot[{f[x], f '[x]}, {x, -Pi, Pi}]

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•Indefinite Integral

Integrate[f[x], x]

-Cos[x]

•Definite Integral

Integrate[f[x], {x, 0, Pi/2}]

1

•Taylor Expansion

Series[Exp[x], {x, 0, 10}]

1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + x^6/720 + x^7/5040 + x^8/40320 + x^9/362880 + x^10/3628800 + O[x]^11

Series[Log[x], {x, 1, 10}]

(x - 1) - 1/2 (x - 1)^2 + 1/3 (x - 1)^3 - 1/4 (x - 1)^4 + 1/5 (x - 1)^5 - 1/6 (x - 1)^6 + 1/7 (x - 1)^7 - 1/8 (x - 1)^8 + 1/9 (x - 1)^9 - 1/10 (x - 1)^10 + O[x - 1]^11

•Differential Equation

DSolve[y '[x] == y[x], y[x], x]

{{y[x] -> e^x C[1]}}

•Applications

•Tangent

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

f[x_] = x^3 - x

-x + x^3

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.

g[x_, a_] = f '[a] (x - a) + f[a]

-a + a^3 + (-1 + 3 a^2) (-a + x)

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

Plot[ {f[x], g[x, 1]}, {x, -2, 2}]

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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.

Solve[f[x] == g[x, 1], x]

{{x -> -2}, {x -> 1}, {x -> 1}}

Plot[ {f[x], g[x, 1]}, {x, -3, 3/2}]

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•Area

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

∫ _ (-2)^1 (f[x] - g[x, 1]) d x

27/4

•Exercises

•Problem 1

Let f(x)=x^4 - 16 x^2. 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)