Bihar Board Class 12 (Commerce) Application of Derivatives — practice questions
90 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice Bihar Board Class 12 (Commerce) Application of Derivatives in the app →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:For a particle moving along x(t) = t³ − 3t² + 5, the velocity at t = 2 is:A disc has radius r and area πr². Compute dA/dr when r = 6 cm.If y = f(x), what does the value of dy/dx at x = x0 represent?If x = f(t) and y = g(t), then by the chain rule dy/dx equals (dx/dt not zero)An edge of a cube grows at 2 cm/s. How fast is the volume increasing when the edge is 5 cm?A balloon's radius grows 3 cm/s. How quickly does its surface area increase at r = 4 cm?A ripple's edge moves at 5 cm/s. At radius 8 cm, the enclosed area increases at what rate?A rectangle has length x falling 2 cm/min and width y rising 3 cm/min. Find the area's rate at x = 8, y = 5.Producing x items costs 3x² + 5x + 200 rupees. Find the marginal cost at 10 items.Selling x items brings R(x) = 5x² + 20x + 10 rupees. Marginal revenue at 4 items?In economics, marginal cost is best described asA 5 m ladder's foot slides from the wall at 3 m/s. How fast does the top fall when the foot is 3 m away?A bubble's volume grows at 200π cm³/s. How fast does its radius grow when r = 5 cm?The region inside a circle expands at 6π cm²/s. Find the circumference's growth rate at r = 3 cm.On the real line, how does 5x + 2 behave?If f is continuous on [a, b] and f'(x) > 0 for every x in (a, b), then f isIf f'(x) = 0 for every x in (a, b) and f is continuous on [a, b], then f isMoving left to right, where does the parabola y = x² fall?On which interval is f(x) = cos x strictly decreasing?On what interval does x² − 6x + 1 fall as x grows?Over which interval does the cubic 2x³ − 9x² + 12x + 5 lose height?Which set of x values makes x³ − 3x strictly decreasing?The cubic x³ has zero slope at the origin. Over all of R it isFor which values of x does eˣ − x rise?For positive x, over what range does x − ln x climb?Over which part of (0, π/2) does sin 3x climb?Find the smallest a for which f(x) = x² + ax + 1 is increasing on [1, ∞).x + 1/x slopes downward on which interval?f is strictly increasing on I if x1 < x2 in I always impliesDescribe y = −2x + 7 as x moves along R.How does 2x − cos x behave over all real numbers?Points where a curve switches between rising and falling are calledA critical point of a function f is a point c whereIf a differentiable f has a local extremum at an interior point c, thenWhen f'(c) = 0 and f''(c) > 0, the point c is aIf f' changes sign from positive to negative through a critical point c, then c is a point ofIf f' does not change sign as x passes through a critical point c, then c isFind the greatest value attained by −x² + 4x + 1.The function f(x) = x³ − 12x has a local maximum at x =Classify x = 0 for the quartic f(x) = x⁴, where f'(0) = 0.Two positive numbers add up to 20. What is the largest possible product?A 60 m fence encloses a rectangle. What is the greatest possible area?For a fixed volume, a closed cylinder has least total surface area when its height equalsDetermine the peak value of sin x + cos x over all real x.On [1, 5], the cubic 2x³ − 15x² + 36x + 1 has what least value?For x > 0, how small can x + 4/x get?A rectangle has area 64 m². What is the smallest possible perimeter?Profit is P(x) = −x² + 40x − 150. How many units give the maximum profit?Revenue is R(x) = 50x − 0.5x². What is the maximum revenue?The absolute extrema of a differentiable function on [a, b] occur only atThe maxima and minima of a function are known collectively as itsIf x* is a global maximum point of f, then f(x*) is ____ f(x) for every x in the domain.For a differentiable function, local extrema in the interior of the domain occur atFinding global maxima and minima is the central goal ofThe set of all real numbers hasTo find the maximum of a piecewise-defined function, one shouldA tangent at point A of a curve can be viewed as the limit of which lines as B approaches A?What is the name for the single point shared by a tangent and its curve?For a curve y = f(x), the slope of the tangent at x = x0 isThe normal to a curve at a point is the line that is ____ to the tangent there.If the tangent to a curve at a point has slope 4, the normal there has slopeGive the tangent line to the parabola y = x² touching it at (1, 1).At which point does y = x² − 4x + 5 have a horizontal tangent?At which point of y = x³ is the tangent parallel to the line y = 12x + 1?Write the line touching y = √x at the point (4, 2).The curves y = x and y = x² meet at (1, 1). If θ is the angle between them, tan θ equalsTwo curves are orthogonal at a point if their tangent lines there areUsing the tangent line approximation of √x near 25, √25.2 is aboutUsing differentials, (1.02)³ is approximatelyUsing linear approximation, the cube root of 8.12 is aboutA square's side is measured as 10 cm with error 0.1 cm. The approximate error in its area isLeibniz defined a tangent as the line throughWater enters a cylinder of base area 10 cm² at 5 cm³/s. How fast does the level rise?A particle moves with s = t³ − 6t² + 9t. At what time t > 0 does it first come to rest?A particle has s = 2t³ − 9t² + 12t. What is its acceleration at t = 1?The velocity of a moving particle is the derivative of itsA square's side grows at 2 cm/s. How fast is its area growing when the side is 7 cm?At radius 3 cm, growing 0.5 cm/s, how quickly is a sphere's volume increasing?A 2 m man walks at 5 km/h away from a 6 m lamp post. How fast does his shadow's tip move?What is the rate of change of a sphere's volume with respect to its radius at r = 10 cm?A rectangle is inscribed in a circle of radius 5 cm. What is its greatest possible area?For x + y = 10 with x, y > 0, the product x²y is greatest when x =Where does x·e^(−x) peak? Give the x-coordinate.A square tile has side x. At what rate does its area change with x when x = 4?