JEE Main Application of Derivatives — practice questions
52 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice JEE Main Application of Derivatives in the app →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 = 0.1If $y = f(x)$, then $\dfrac{dy}{dx}$ represents the:The rate of change of the area of a circle ($A = \pi r^2$) with respect to its radius, when $r = 5$, is:A function $f$ is strictly increasing on an interval if, throughout it:A function $f$ is strictly decreasing on an interval if, throughout it:The slope of the tangent to the curve $y = f(x)$ at the point $x = a$ is:The slope of the normal to the curve $y = f(x)$ at $x = a$ is:At a critical point (a candidate for a local maximum or minimum) of a differentiable function $f$:The function $f(x) = x^2$ is strictly increasing for:The rate of change of the volume of a sphere $V = \tfrac{4}{3}\pi r^3$ with respect to its radius $r$ is:For $f(x) = x^2 - 4x + 3$, the critical point occurs at:The slope of the tangent to $y = x^2$ at the point $(1,1)$ is:By the second derivative test, if $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a point of:By the second derivative test, at a point of local maximum the value of $f''(x)$ is:A function $f$ is increasing on an interval if, on that interval:The slope of the tangent to $y = x^2$ at $x = 3$ is:By the second derivative test, a critical point $c$ of $f$ with $f''(c) > 0$ is a:At a local maximum of a differentiable function $f$ on its domain:A function f(x) is INCREASING on an interval if for all x in the interval:The tangent to y = x² at the point (1, 1) has slope:Rolle's theorem requires that on [a, b], in addition to continuity + differentiability:At a critical point c where f'(c) = 0, if f''(c) > 0, then c is a:To find the dimensions of a rectangle with maximum area + perimeter 20, set up:The critical points of f(x) = x³ − 3x + 1 are found by solving f'(x) = 0:A balloon is inflated so its volume V = (4/3)πr³ increases at 8π cm³/s. When r = 2 cm, the rate of change of rFor a particle moving along x(t) = t³ − 3t² + 5, the velocity at t = 2 is:At the optimal production level (maximum profit), in marginal analysis: