TS EAPCET (Engineering) Application of Derivatives — practice questions
84 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice TS 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 = 0.1A particle moves so that its position is s = t³ - 6t. Find its velocity at t = 2.A particle has position s = t³ - 6t. What is its acceleration at t = 2?At what rate does the area of a circle change with respect to its radius when r = 5?At which x is the tangent to y = x² parallel to the line y = 8x + 1?At which values of x does the curve y = x³ - 3x have a horizontal tangent?On which interval is f(x) = x² - 4x decreasing?Where is f(x) = x³ - 3x increasing?Find the critical point of f(x) = x² - 6x + 5.If f'(c) = 0 and f''(c) > 0, then at x = c the function has aIf f'(c) = 0 and f''(c) = 0, what does the second derivative test tell us?If f' changes from positive to negative as x increases through c, then f has at c aFermat's theorem says that at an interior local extremum of a differentiable functionA continuous function on a closed interval must, by the extreme value theorem, haveWhat is the absolute maximum of f(x) = x² on the interval [-3, 2]?Two positive numbers add up to 20. What is the greatest possible product?A farmer has 60 m of fencing for a rectangular plot. What is the largest area it can enclose, in square metresFor positive x, what is the least value of x + 1/x?Squares of side x are cut from the corners of a 12 cm square sheet to make an open box. What is the greatest vTwo non-negative numbers add up to 10. What is the least possible value of the sum of their squares?A 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 out?Air is pumped into a spherical balloon at 100 cm³/s. How fast is the radius growing when r = 5 cm?The radius of a circle grows at 2 cm/s. How fast is the area increasing when the radius is 5 cm?After naming the known and wanted rates, what is the next step?Why substitute values only after differentiating in related rates?Which rule is mainly needed when an equation is differentiated with respect to time in related rates?If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), Rolle's theorem givesFor f(x) = x² - 4x + 3 on [1, 3], at which c does Rolle's theorem give f'(c) = 0?Rolle's theorem cannot be applied to f(x) = |x| on [-1, 1] becauseFor f(x) = x² on [2, 6], at which c does the mean value theorem hold?For f(x) = x³ on [0, 2], the mean value theorem gives which value of c in (0, 2)?By the mean value theorem, an interior tangent is parallel to theA car covers 120 km in 2 hours. By the mean value theorem, at some instant its speed equalsWhich 12th century Indian mathematician stated an early form of what is now called Rolle's theorem?Which of these functions is not differentiable at x = 0?In kinematics, the second derivative of position with respect to time givesA particle moves with s = t³ - 3t². At what positive time is it momentarily at rest?Where can the global maximum of a function on a domain occur?Which of these sets has a minimum but no maximum?Using the tangent line as a linear approximation, estimate √36.1.At what x does f(x) = x³ - 3x² have a point of inflection?