The area of the region enclosed by the curve \(y=x^3\) and its tangent at the point (−1,−1) is
4
27
\(\frac{4}{27}\)
\(\frac{27}{4}\)
The Correct Option is D
Solution and Explanation
The equation of the tangent is:
\( y + 1 = 3(x + 1) \)
Simplifying:
\( y = 3x + 2 \)
The point of intersection of the tangent with the curve is: \( (2, 8) \)
The area bounded by the curve and the tangent is:
\( \text{Area} = \int_{-1}^{2} (3x + 2 - x^3) \, dx \)
Evaluate the integral:
\( \int_{-1}^{2} (3x + 2 - x^3) \, dx = \frac{27}{4} \)
Thus, the area is:
\( \frac{27}{4} \)
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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