AP EAPCET (Engineering) Application of Derivatives — practice questions
75 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice AP EAPCET (Engineering) 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 curve y = 2x^2 - 3x has what tangent slope at x = 1?The tangent to y = x^3 - 3x is parallel to the x-axis at which values of x?The function f(x) = x^2 - 4x is increasing for which values of x?The function f(x) = 3 - 2x behaves how on the real line?The function f(x) = x^3 is what over all real numbers?On which interval is f(x) = x^3 - 3x decreasing?In which interval is sin x an increasing function?The function cos x is decreasing on which interval?If f'(x) > 0 at every point of an interval, then f is what on it?What is the minimum value of f(x) = x^2 - 6x + 10?At which x does f(x) = x^2 - 4x + 3 have a local minimum?For positive x, what is the least value of x + 1/x?Two positive numbers add up to 20. What is the largest possible product?A rectangle has perimeter 40 cm. What is its greatest possible area?Two positive numbers sum to 12. What is the least value of the sum of their squares?Local extrema inside the domain of a differentiable function occur where the derivative is what?What happens to f(x) = x^3 at its critical point x = 0?If f'(c) = 0 and f''(c) < 0, then f has what at c?A function continuous on a closed interval is guaranteed to attain what there?What does f(x) = |x| have at x = 0?What is the absolute maximum of f(x) = 2x - x^2 on [0, 3]?A circle's radius grows at 2 cm/s. How fast is its area growing when r = 5 cm?A cube's edge grows at 3 cm/s. How fast does its volume grow when the edge is 2 cm?A 10 m ladder's foot slides out at 3 m/s. How fast does its top fall when the foot is 6 m out?A balloon's volume grows at 100 cu cm/s. How fast does its radius grow at r = 5 cm?A circle's radius changes at 0.5 cm/s. How fast does its circumference change?In a related rates problem, once the equation is built, one differentiates with respect to what?A point moves on y = x^2 with x increasing at 3 units/s. How fast does y change at x = 2?The derivative of displacement with respect to time gives which quantity?Using the tangent line at x = 25, what is the linear approximation of sqrt(26)?If y = x^2 and x changes from 3 to 3.01, what is the approximate change dy?