KEAM Application of Derivatives — practice questions
50 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice KEAM 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:$y = x^3 - 6x^2 + x + 3$. The tangent is parallel to the line $y = x + 5$ at points where $x$ equals: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$):**Lagrange's Mean Value Theorem (MVT)** states: for $f$ continuous on $[a,b]$, differentiable on $(a,b)$, ther$f(x) = 2x^3 - 3x^2 - 12x + 5$. The **local maximum value** occurs at:A 13-m ladder leans against a wall. Its top slides down at 0.5 m/s. When the foot is 5 m from the wall, the foThe 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 = $:A car moves so that its position $s(t) = 3t^3 - 12t^2 + 5$ (m, t in s). The time at which the car comes to **mTangent to the curve $y = qrt x + qrt y = c$ (where $c > 0$) at any point on it makes equal intercepts on thThe 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:Find $ in 31°$ using approximation (given $ in 30° = 0.5$, $\cos 30° = qrt 3/2 \approx 0.866$, $1° = 0.01745$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 = 0.1