Zadanie 2e
For (x − 3)2 + y2 = 9, (i) find dy/dx in terms of x and y; (ii) hence or otherwise find all x-values where the tangent gradient is undefined.
Implicit derivatives and related rates
Differentiate implicit relationships, link rates through the chain rule and solve constrained optimization problems including endpoints.
Tę umiejętność sprawdzają 32 zadania wzorowane na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 32 Otwórz w wyszukiwarce z filtrami
Zadanie 6
A rectangular light spot maintains area 240 cm2 as its length L and width W change, so LW = 240. At one instant L=12 cm and L is decreasing at 3 cm/s. Find the rate of increase of W.
- Materiał graficzny
Zadanie 8a
Find dy/dx in terms of x and y.
Zadanie 8b
At P the y-coordinate increases at 4 units per second. Find the instantaneous rate of change of the x-coordinate.
Zadanie 8b
Find dr/dt when h=3 cm.
Zadanie 12b
Show that y″(0)=4.
Zadanie 2e
Consider the lower arc of the circle (x − 2)2 + (y + 1)2 = R2, where R > 0, 2 − R < x < 2 + R and y < −1. (i) Show that . (ii) Hence show that the curvature is constant along this arc.
Zadanie 9b
At an instant when θ = 0.2 radians, the angle is increasing at 6 radians per hour. Find the aircraft’s speed.
Zadanie 12a
Use implicit differentiation to establish .
- Materiał graficzny
Zadanie 12b
Hence find the full set of x-coordinates on C in the form , with positive integer a.
Zadanie 12c
Find the coordinates of A.
Zadanie 12d
A line y = mx, where m is an integer, is tangent to C. Determine every possible m.
Zadanie 9
Find dx/dt when the triangle area is 81 cm².
Zadanie 12b
(i) Show that (x2 + xy)y′′ = 2x + y − x(y′)2 − (x + 17y)y′. (ii) Find y″ at x=1.
Zadanie 6
A soft hemispherical dome expands while retaining its hemispherical shape. Material enters at a constant rate of 8 cm³ s⁻¹. Its volume is V = (2/3)πr3. Find the rate at which its radius increases when V=36 cm³.
Zadanie 11a
Show that dy/dx = (2x − ex+y)/(ex+y − 2y) wherever the denominator is nonzero.
Zadanie 11b
(i) Show that the horizontal-tangent x-coordinates satisfy 2x2 + [ln(2x)]2 − 2xln(2x) − 2x − 1 = 0, with x>0. (ii) Find P and Q.
Zadanie 11c
Using symmetry, state the coordinates of the two points where the tangent is vertical.
Zadanie 11d
Find the point on C whose tangent has gradient −1.
Zadanie 11c
Use y′ = (1 − y2)3/2. (i) By implicit differentiation, show that y′′ = −3y(1 − y2)2. (ii) Hence express y′′′ in terms of y.
Zadanie 2h
Show that the maximum of g occurs at x=λS for a constant λ independent of S, and determine λ numerically.
Zadanie 6
A rhombus keeps an acute interior angle of π/3 while each side x cm grows at 3 cm/s. Find the rate of increase of its area when cm.
Zadanie 12b
(i) State y′(0). (ii) Show that y″(0)=18.
Zadanie 2d
Show that E_a and H_b intersect orthogonally at their first-quadrant intersection M. You may use xM2 = (a2 + 9)(9 − b2)/9 and yM2 = a2b2/9.
Zadanie 1d
Now restrict y2 = (x − 1)3 − 4(x − 1) to its right component x≥3. (i) Show that for x>3, . (ii) Hence explain why neither open branch x>3 has a local maximum or minimum.
Zadanie 1e
Now restrict y2 = (x − 1)3 − 4(x − 1) to its right component x≥3. The two inflection points on this right component share an x-coordinate. Find that coordinate in exact radical form.
Zadanie 1f
A rational point has two rational coordinates. On C : y2 = x3 − 3x + 4, let P=(0,2). The tangent at P meets C again at Q; let S=(0,−2). (i) Find the tangent equation at P. (ii) Hence find Q, expressing its coordinates as fractions.
Zadanie 10e
Find dh/dt when half the water capacity has been filled.
Zadanie 12b
Show that d2P/dt2 = k2(P − H)(1 − P/N)[1 + (H − 2P)/N].
Zadanie 12c
Using P′′ = k2(P − H)(1 − P/N)[1 + (H − 2P)/N], show that the rate of adoption is greatest at P = (N + H)/2, and justify the maximum.
Zadanie 7
The curve x2y2 + y6 = 9x2 is restricted to x>0 and −3<y<3. Show that it has no local maximum or local minimum points.
Zadanie 8
A widening vessel is filled at a constant rate of 3 cm³/s. For liquid depth h cm, its volume is V = π(2h2 + h3/6), for 0≤h≤8. Find how fast the depth is increasing when the vessel contains 100 cm³.