Question

Statement I-Geometrically, derivative of a function is the slope of the tangent to the corresponding curve at a point.
Statement II- Geometrically, indefinite integral of a function represents a family of curves parallel to each other.
a)Statement I is true; statement II is true; statement II is a correct explanation for statement I.
b)Statement I is true; statement II is true; statement II is not a correct explanation for statement I.
c)Statement I is true; statement II is false.
d)Statement I is false; statement II is true.

Answer 1

Hint: Derivation of function of variable x which is the rate at the value of function changes with respect to the change of variable. Integration is a method of finding the area to the x axis from the curve.

Complete step-by-step answer:
We know the Geometrical meaning of the derivative of a function at a point is the slope of the tangent to the corresponding curve at a point.
Let there is a curve, \[y = f\left( x \right)\]
Slope of the tangent $ = \tan \theta $ $ = \dfrac{{dy}}{{dx}}$ at (x,y)
And ,
Geometrically, the indefinite integral represents a family of curves that place each other having parallel tangents of intersection of the curves of the lines perpendicular to the axis, the variable of integration.
As integral is the antiderivative of the function.
$\int {\dfrac{{df\left( x \right)}}{{dx}}dx = f\left( x \right)} + C$
Here C represents the family of curves.
Both statements are true.
But both derivative and integral are different, so statement II is not a correct explanation for statement 1.
Correct option must be b).

Note: Don’t confuse between derivative and integral, always remember that derivative is used for a point whereas integral is never used for a point instead it is used for integral of a function over an interval.