TS EAMCET Application of Derivatives — practice questions
85 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice TS EAMCET Application of Derivatives in the app →The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:A function $f$ is **strictly increasing** on an interval if, on that interval:At a **local maximum** of a differentiable function $f$:The slope of the tangent to $y = x^3 - 3x + 1$ at $x = 2$ is:Water is poured at $36\,\text{m}^3/\text{s}$ into a cylindrical vessel of base radius 3 m. The rate at which wIf $f(x) = x^2 - 4x + 3$, the **interval** on which $f$ is decreasing is:Find the approximate value of $ qrt{25.5}$ using differentials (given $ qrt{25} = 5$):$f(x) = 2x^3 - 3x^2 - 12x + 5$. The **local maximum value** occurs at:The function $f(x) = x^3 - 3x + 1$ has critical points at:Find the **maximum value** of $f(x) = in x + \cos x$ for $x \in [0, 2\pi]$.A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:Find the **shortest distance** from origin to the curve $xy = 4$ (in the first quadrant):If $y = a \ln x + b x^2 + x$ has extreme values at $x = 1$ and $x = 2$, then $(a, b)$ is:The function $f(x) = \dfrac{x}{1 + x^2}$ attains its maximum value on $[0, \infty)$ at:If $f(x) = x^x$ for $x > 0$, then $f$ has a minimum at $x = $:The volume of a sphere is increasing at $8\pi$ cm³/s. The rate of change of surface area when the radius is 2 If the tangent to the curve $y = e^{2x}$ at point $(a, e^{2a})$ passes through origin, then $a$ equals:A wire of length $L$ is bent to form a rectangle. The **maximum area** enclosed is:Slope of tangent to y = f(x) at point (a, f(a)) is:Slope of normal line to y = f(x) at x = a (if f'(a) ≠ 0):At a stationary point of f, derivative f' equals:Function f is INCREASING on interval (a, b) if:Local maximum of f at x = c (if f'(c) = 0):Mean Value Theorem says that for continuous f on [a, b] differentiable on (a, b), there exists c ∈ (a, b) withFor y = x³ - 3x² + 2, find critical points:Point of inflection of curve y = f(x) is where:For y = sin x, find dy/dx at x = π/3:Approximation by differentials: Δy ≈ f'(x) Δx. Using this, sqrt(25.5) is approximately:Rate of change of area of a circle (A = πr²) with respect to radius:If two functions f and g satisfy f'(x) = g'(x) for all x on an interval, then:Rate of change of volume V = (4/3)πr³ of a sphere with radius:Linear approximation of f(x) near x = a:A spherical balloon's volume increases at 30 cm³/s. At r = 5 cm, how fast is the radius increasing?Find equation of tangent to y = x² - 1 at point (2, 3):Maximum area of a rectangle inscribed in a semicircle of radius R (base on diameter):Use L'Hôpital's rule: lim_(x→0) (sin x - x)/x³ =For f(x) = x² ln x on (0, ∞), find x where f has local minimum:A 5 m ladder slides down a wall. Bottom moves outward at 1 m/s. Speed of top when bottom is 3 m from wall:For y = e^x − x, find values where tangent is horizontal:For f(x) = x³ - 3x² + 4, the function is increasing on intervals:Find minimum value of f(x) = (sin x)² + (cos x)² + 2 (sin x)(cos x):Volume of cylinder inscribed in sphere of radius R, expressed using h (height of cylinder):Concavity test: f(x) = e^x is:Find the rate at which the surface area of a balloon (sphere) is changing when radius is 10 cm and dr/dt = 0.1The tangent to y = x^2 + 4x - 16 is parallel to the x-axis at x equal toThe function f(x) = x^2 - 6x + 5 is strictly decreasing on the intervalThe function f(x) = 2x^3 - 9x^2 + 12x + 1 is increasing onThe minimum value of f(x) = x^2 - 4x + 7 isFor f(x) = x^2 - 4x + 3 on [1, 3], the point c given by Rolle's theorem isFor f(x) = x^3 on [0, 2], the value of c in the mean value theorem isRolle's theorem cannot be applied to f(x) = |x| on [-1, 1] because f fails to be differentiable atA 10 m ladder's base slides from a wall at 3 m/s. How fast does the top fall when the base is 6 m away?The radius of a circular oil patch grows at 2 cm/s. How fast is its area increasing when the radius is 5 cm?Each edge of a cube expands at 3 cm/s. At the instant the edge is 2 cm, the volume increases at the rate ofA spherical balloon's radius grows at 1 cm/s. Its surface area increases at what rate when the radius is 10 cmIf f'(c) = 0 and f''(c) < 0 at an interior point c, the second derivative test says f hasFor x greater than 0, the least value of x + 1/x isA rectangle has a perimeter of 40 cm. Its greatest possible area isTwo positive numbers add up to 16. The least possible value of the sum of their squares isAn open box is made from an 18 cm square sheet by cutting corner squares. Which cut side gives maximum volume?For f(x) = x^3, the stationary point x = 0 isAt x = 1/2, the tangent to y = x^2 makes what angle with the positive x-axis?A particle moves with s = t^3 - 6t^2 + 9t. Its acceleration becomes zero at t equal toA body's distance is s = 5t^2 + 2t metres. Its velocity at t = 3 s isThe rate of change of a cube's volume with respect to its edge a at a = 3 isWhich condition is part of the hypothesis of Rolle's theorem on [a, b]?Rolle's theorem guarantees a point c inside (a, b) at whichLagrange's mean value theorem asserts a point c in (a, b) such that f'(c) equalsGeometrically, the mean value theorem says some tangent to the curve isMichel Rolle's proof of 1691 covered only which kind of function?Which Indian mathematician stated an early form of Rolle's theorem in the 12th century?Which Kerala School scholar described a special case of the mean value theorem for the sine?A function continuous on a closed and bounded interval always attainsBy Fermat's theorem, an interior local extremum of a differentiable function occurs where f'(x) isIn related rates, after writing the linking equation, the next step is toThe side of a square plate grows at 2 cm/s. Its area rises at what rate when the side is 5 cm?A sphere's radius changes with time. At what radius do surface area and volume change at equal numerical ratesThe point of inflection of f(x) = x^3 - 6x^2 + 9x lies at x equal toThe curve y = ln x on (0, infinity) isOn the interval (0, π), the function cos x isUsing differentials, the approximate value of √25.1 isFor y = x^2, x changes from 10 to 10.1. The differential dy isThe largest area of a rectangle that can be inscribed in a circle of radius r isAmong rectangles of area 100 cm^2, the least possible perimeter isThe function f(x) = |x - 2| has its minimum at x = 2, where its derivative