Question

Вычислите площадь фигуры, ограниченной графиком функции y=1,5x^2+3, касательной к этому графику в точке с абсцисой x=2 и прямой x=0. Желательно с рисунком, заранее спасибо.

Answer 1

РЕШЕНИЕ
Уравнение касательной по формуле
Y = k*x + b = y'(Xo)*(x -Xo)+ y(Xo)
Уравнение производной
y'(x) 2*1.5*x = 3*x = k.
Вычисляем в точке касания.
y'(2) = 6,  y(2) = 1.5*4 = 6.
Уравнение касательной
Y = 6*(x - 2) + 6 = 6*x - 3 
Находим пределы интегрирования - решаем уравнение:
1,5*x² + 3 = 6*x - 3
1.5*x² - 6*x + 6 = 0 
a = x1=x2 = 2 
b = 0 - дано
Площадь - интеграл разности функций = 1.5*x²+3 -(6*x - 3)
$S= \int\limits^2_0 {6-6x+ \frac{3}{2}x^2 } \, dx= \frac{6x}{1}- \frac{6x^2}{2}+ \frac{3/2x^3}{3}$
Вычисляем при а - S(2) = 4
Вычисляем при b - S(0) = 0
S = 4 - площадь - ОТВЕТ
рисунок к задаче в приложении.

Answer 2

Calculating the Area of the Bounded Figure

To calculate the area of the figure bounded by the graph of the function y=1.5x^2+3, the tangent to this graph at the point with x=2, and the line x=0, we can use the definite integral of the function.

The area can be calculated by finding the definite integral of the absolute difference between the two functions over the interval [0, 2].

The formula for the area A bounded by the curves f(x) and g(x) over the interval [a, b] is given by:

A = ∫[a, b] |f(x) - g(x)| dx

In this case, the functions are y=1.5x^2+3 and the tangent line at x=2, and the interval is [0, 2].

Calculation Steps

1. First, we need to find the equation of the tangent line at x=2. 2. Then, we'll calculate the definite integral of the absolute difference between the function y=1.5x^2+3 and the tangent line over the interval [0, 2].

Equation of the Tangent Line at x=2

The equation of the tangent line at x=2 can be found using the derivative of the function y=1.5x^2+3 and the point (2, f(2)).

The derivative of the function y=1.5x^2+3 is y'=3x.

Using the point-slope form of a line, the equation of the tangent line is given by:

y - f(2) = y'(2)(x - 2)

Calculating the Definite Integral

After finding the equation of the tangent line, we can calculate the definite integral of the absolute difference between the function y=1.5x^2+3 and the tangent line over the interval [0, 2].

Let's calculate the definite integral to find the area of the bounded figure.

Area Calculation

The area A bounded by the curves y=1.5x^2+3 and the tangent line at x=2 over the interval [0, 2] can be calculated using the definite integral:

A = ∫[0, 2] |(1.5x^2+3) - (3x)| dx

Now, let's calculate the definite integral to find the area of the bounded figure.