68 free MCQs on Application of Derivatives, each with its own worked answer and explanation. Use derivatives to study rate of change, increasing and decreasing functions, tangents and normals, and maxima and minima, with classic optimization problems.
Take the timed Application of Derivatives chapterwise test →68 practice questions on Application of Derivatives, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Application of Derivatives notes.
The tangent at a point touches the curve with slope f'(a); the normal is the line perpendicular to the tangent at that same point, with slope -1/f'(a) - together they describe the curve's local direction and the line "straight into" the curve.
Q1.
If f'(x) > 0 for all x in an interval, the function f is:
Q2.
If f'(x) < 0 for all x in an interval, the function f is:
Q3.
A critical point of a function f is a point where:
Q4.
The slope of the tangent to the curve y = f(x) at point (x<sub>1</sub>, y<sub>1</sub>) is given by:
Q5.
If the slope of the tangent at a point is m, the slope of the normal at that point is:
Q6.
At a point where the tangent is horizontal, f'(x) equals:
Q7.
By the second derivative test, if f'(c) = 0 and f''(c) > 0, then x = c is a:
Q8.
By the second derivative test, if f'(c) = 0 and f''(c) < 0, then x = c is a:
Q9.
For the function f(x) = x<sup>2</sup>, the value of f'(x) is:
Q10.
If Area of a circle A = pi r<sup>2</sup>, then dA/dt in terms of dr/dt is:
Q11.
Rolle's theorem requires which of the following conditions on [a, b]?
Q12.
The Mean Value Theorem guarantees a point c in (a, b) where f'(c) equals:
Q13.
Using the approximation formula, f(x + dx) is approximately equal to:
Q14.
The derivative dy/dx represents the ___ of a function:
Q15.
The slope of the tangent to a curve y = f(x) at a point is given by:
Q16.
At a point of local maximum or minimum, the first derivative f′(x) is:
Q17.
A function is increasing on an interval where its derivative f′(x) is:
Q18.
A function is decreasing on an interval where f′(x) is:
Q19.
The second derivative f″(x) is used to determine the ___ of a curve:
Q20.
Points at which f′(x) = 0 are called ___ points:
Q21.
Find the equation of the tangent to the curve y = x<sup>2</sup> at the point (2, 4).
Q22.
Find the equation of the normal to the curve y = x<sup>3</sup> at the point (1, 1).
Q23.
Find the interval in which f(x) = x<sup>2</sup> - 4x + 3 is strictly decreasing.
Q24.
Find the critical points of f(x) = x<sup>3</sup> - 3x.
Q25.
For f(x) = x<sup>3</sup> - 6x<sup>2</sup> + 9x + 1, the local maximum occurs at:
Q26.
For f(x) = x<sup>3</sup> - 6x<sup>2</sup> + 9x + 1, the local minimum value is:
Q27.
Two positive numbers have sum 16. Find the maximum value of their product.
Q28.
Using differentials, approximate the value of the square root of 25.2.
Q29.
The radius of a sphere is increasing at the rate of 2 cm/s. Find the rate of increase of its volume when the radius is 5 cm.
Q30.
Find the point on the curve y = x<sup>2</sup> where the tangent is parallel to the line y = 4x - 5.
Q31.
Verify Rolle's theorem for f(x) = x<sup>2</sup> - 4x + 3 on [1, 3]. The value of c satisfying f'(c) = 0 is:
Q32.
Find c using the Mean Value Theorem for f(x) = x<sup>2</sup> on the interval [1, 4].
Q33.
A balloon's volume increases such that its radius increases at 3 cm/s. Find the rate of change of surface area when r = 4 cm. (Surface area = 4 pi r<sup>2</sup>)
Q34.
By the second-derivative test, if f′(x) = 0 and f″(x) > 0, the point is a local:
Q35.
If f′(x) = 0 and f″(x) < 0, the point is a local:
Q36.
The equation of the tangent to y = f(x) at (x₁, y₁) is y − y₁ =:
Q37.
The rate of change of the area A = πr² of a circle with respect to its radius r is:
Q38.
For a sphere of volume V = (4/3)πr³, the rate of change of volume with respect to r is:
Q39.
The slope of the normal to a curve is the ___ of the tangent’s slope:
Q40.
For a small change dx, the approximate change in y is dy =:
Q41.
Show that among all rectangles with a given perimeter of 40 units, the one with maximum area is a square. What is the maximum area?
Q42.
An open box is to be made from a square sheet of side 12 cm by cutting equal squares of side x from each corner and folding up the sides. Find x that maximizes the volume.
Q43.
Find two positive numbers x and y such that x + y = 10 and x<sup>2</sup> + y<sup>2</sup> is minimum. Find this minimum value.
Q44.
Among all closed cylindrical cans of a fixed volume V, the surface area is minimized when the height h and radius r satisfy:
Q45.
A man of height 2 m walks away from a lamp post of height 6 m at a speed of 1.5 m/s. Find the rate at which the length of his shadow increases.
Q46.
Find the maximum value of the function f(x) = -2x<sup>3</sup> + 3x<sup>2</sup> + 12x - 5 on the interval [-2, 3].
Q47.
A wire of length 36 m is to be cut into two pieces. One piece (length 4x) is bent into a square of side x, and the other into a circle of radius r. To minimize the total area enclosed, what relation must hold between x and r?
Q48.
Verify the Mean Value Theorem for f(x) = x<sup>3</sup> - 5x<sup>2</sup> - 3x on [1, 3]. Find the value of c in (1, 3) satisfying the theorem.
Q49.
The sum of the surface areas of a cube and a sphere is constant. Show that the sum of their volumes is minimum when the edge of the cube equals the diameter of the sphere. If the cube's edge is a and sphere's radius is r at the minimum, what is the relationship?
Q50.
Find the maximum area of an isosceles triangle inscribed in a circle of radius r, with the triangle's apex at the top of a vertical diameter.
Q51.
A point moves along the curve y = x<sup>3</sup> - 3x. Find the points where the tangent to the curve is parallel to the x-axis, and classify them.
Q54.
Among all rectangles of a fixed perimeter, the area is greatest when the rectangle is a:
Q55.
For f(x) = x³ − 3x, the critical points occur at x =:
Q56.
The point where a curve changes its concavity is called a point of:
Q57.
If f′(x) > 0 for every x in an interval, the function is:
Q58.
The derivative of the position of a particle with respect to time gives its:
Q59.
The second derivative of position with respect to time gives the:
Q60.
A rectangle of fixed perimeter 20 has maximum area when each side equals:
Q61.
The value of x > 0 at which x<sup>1/x</sup> attains its maximum is:
Q62.
The function f(x) = x³ − 3x is decreasing on the interval:
Q63.
The equation of the tangent to y = x² at the point (1, 1) is:
Q64.
Two numbers have a sum of 20. The maximum value of their product is:
Q65.
A spherical balloon's volume increases at 100 cm³/s. When the radius is 5 cm, the radius increases at:
Q66.
Using differentials, the approximate value of √26 is:
Q68.
The point on the parabola y² = 4x nearest to (2, 1) is:
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