Zadanie 1i
Let . At the inflection point x=t, the tangent is y=mx+c, where m = (16b/7)t3. Show that c = d + 9b2/28.
Tangent and normal equations
Determine tangent or normal gradients and equations at specified points using derivatives and line relationships.
Tę umiejętność sprawdza 25 zadań wzorowanych na arkuszach IB z lat 2022–2026.
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Zadanie 10c
The graph meets the vertical axis at P. Prove that the tangent at x=3 contains P.
Zadanie 12d
Find the tangent equation at x=2.
Zadanie 1d
(i) Prove the identity . (ii) Hence give the equation of its oblique asymptote. (iii) Give one point on the curve at which the tangent is parallel to that asymptote.
Zadanie 4b
Hence determine every slope m for which the line is tangent to the parabola.
Zadanie 5a
Find the gradient of Lk in terms of k.
Zadanie 5b
The smaller angle between Lk and the horizontal is π/10. Find all possible k.
Zadanie 5b
The tangent at x = 1 has inclination 35° to the positive horizontal axis. Determine all possible values of a.
Zadanie 12d
Consider two distinct solution curves passing through (−3, p) and (−3, q), where p ≠ q. Use their tangents at x = −3 to prove that every tangent at this x-coordinate, for this family of solutions, passes through a common point. State that point.
Zadanie 1a
(i) Show that the normal to H at x=t has slope −t2/9. (ii) Hence show that its equation is y = −(t2/9)x + t3/9 − 9/t.
Zadanie 1b
(i) Eliminate t to show that every such normal has m<0 and (−m)c2 = 9(m2 − 1)2. (ii) State the set of possible intercepts c.
Zadanie 1c
For which intercepts c is there (i) exactly one distinct normal to H through (0,c); (ii) exactly two distinct normals through (0,c)?
Zadanie 1d
On one pair of axes sketch (i) H; (ii) its two distinct normals passing through (0,1). Mark each point of normal contact. Exact contact coordinates and line equations are not required.
Zadanie 1e
The normal to F at A is y = −t2x + kt3 − k/t. (i) Show that its intersections with F satisfy x2 − k(t − t−3)x − k2/t2 = 0. (ii) Hence, using a sum/product of roots or another valid method, find B.
Zadanie 1f
Prove that angle BCA is a right angle.
Zadanie 9c
Take k=6. Show that the normal at x=−4 is the line L : y = −x/2 + 2.
Zadanie 8b
Hence show that the tangents to f and g at P are perpendicular.
Zadanie 2a
(i) Express the slope of y=kx² using only x,y. (ii) Hence show its orthogonal trajectories satisfy y′=−x/(2y). (iii) Solve this differential equation to show that these trajectories have equation x²+2y²=C, with constant C.
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 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 2b
Show that y=24x−12 is tangent to y=f(x) at A=(2,36).
Zadanie 2c
Sketch y=f(x) and its tangent at A, marking the tangent’s x-intercept.
Zadanie 2d
Assume a≠r. (i) Show that g′(x) = 2(x − r)[(x − a)2 + b2] + 2(x − r)2(x − a). (ii) Hence, or otherwise, prove that the tangent at A meets the x-axis at T = ((a + r)/2, 0).
Zadanie 2f
The figure concerns g(x) = (x + 1)2[(x − a)2 + 4]. The curve touches the x-axis at R=(−1,0). Its tangent at A=(a,64) crosses the x-axis at T=(1,0). Points C₁,C₂ represent respectively the nonreal roots with positive and negative imaginary parts of g(z)=0, drawn on…
- Materiał graficzny
Zadanie 4
The curve y = (3x + 2)ekx has a tangent at x=1 parallel to the straight line y = (7ek)x. Find the rational constant k.