“Application of Derivatives” is crucial because it connects derivative concepts to real problems like finding tangents and normals, estimating function values, optimizing areas/volumes, and proving properties such as monotonicity and roots. In board exams and competitive tests, these problems directly assess both computational skill (differentiation + solving equations) and conceptual clarity (interpretation of derivative signs, linear approximation, and Rolle’s/mean value theorem reasoning).
25
Minutes
20
Questions
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Marking
Q1. Use linear approximation (first-order Taylor expansion) of about to estimate .
Q2. Water is draining from an inverted right circular cone of height m and base radius m. If the volume of water is decreasing at the rate , how fast is the water level falling when the depth of water is m? (Take the outflow rate as .)
Q3. Find the equations of the tangents to the curve which are parallel to the line .
Q4. Let be a real function for which derivatives up to order three may or may not exist at a point . Consider the statements:
A: There exists a function such that and is a point of inflection.
R: For any function , if then .
Both A and R are true.
Both A and R are false.
A is true but R is false.
A is false but R is true.
Q5. Define f(x)=\begin{cases}x^{2}\sin\!\left(\dfrac{1}{x}\right), & x\neq 0,\$$4pt]0, & x=0.\end{cases} Which one of the following is true about at ?
is not differentiable at .
is differentiable at but is not continuous at .
is differentiable at and is continuous at .
is differentiable at but approaches a finite non-zero limit as .
Q6. Find the points on the curve at which the tangent is horizontal.
and
and
and
and
Q7. For the curve , the tangent at a point meets the coordinate axes and forms a triangle. The maximum possible area of such a triangle is
Q8. An open-top rectangular box has a square base of side (m) and height (m). If its volume is , the value of that minimizes the total surface area (base + four sides) is
Q9. Assertion (A): For any real constants , the equation has at most two real roots.
Reason (R): If then for all , so is strictly increasing and hence can have at most one real zero; by Rolle's theorem this implies can have at most two real zeros.
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q10. Assertion (A): If is differentiable on and for all (for some constant ), then is uniformly continuous on .
Reason (R): By the Mean Value Theorem, for any there exists between and with , so is Lipschitz and hence uniformly continuous.
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q11. Using the linear approximation of about , estimate .
Q12. A rectangular sheet of metal of size is made into an open-top box by cutting congruent squares of side from the corners and folding up the sides. For , the value of that maximizes the volume is
Q13. For the function , the maximum value for is
Q14. Assertion (A): For the point is a stationary point of .
Reason (R): A stationary point of a function (when defined in the usual sense) is a point where the derivative equals .
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(A) is true but (R) is false.
(A) is false but (R) is true.
Q15. Points on the parabola are considered. How many points on this parabola are closest to the point ?
One point:
Two points:
Infinitely many points
One point:
...and 5 more challenging questions available in the interactive simulator.