Zadanie 1h
For b<0, show that the real zeros of f″ are 0 and .
Second derivatives and linked graphs
Interpret higher-gradient information and relate the graphs and signs of a function and its first two derivatives.
Tę umiejętność sprawdzają 23 zadania wzorowane na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 23 Otwórz w wyszukiwarce z filtrami
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.
Zadanie 10a
Show that a=1 and b=−6.
Zadanie 12e
Use the curve’s shape to explain whether the Euler estimate at x=1.10 is above or below the exact value.
Zadanie 3b
Find the interval on which its graph is concave up.
Zadanie 9b
The diagram shows two arcs of the associated solution relation. (i) Is the estimate in (a) above or below the upper arc at x = 3? Justify your answer using its curvature. (ii) Explain why forward Euler steps starting at (2, 0) cannot estimate the point on the…
- Materiał graficzny
Zadanie 9c
Find c.
Zadanie 2a
For G, find the second derivative.
Zadanie 12e
Explain why Euler’s method with the specified step underestimates the exact solution at grid points in 0<x<1.
Zadanie 12d
Explain using concavity why the exact value at 1.05 exceeds the one-step Euler estimate. Use and one forward Euler step from x=1,y=3 of length 0.05.
Zadanie 12e
On , the solution has an inflexion point P. Sketch an appropriate derivative graph and find the x-coordinate of P. Use the explicit solution .
Zadanie 12f
The inflexion point P lies in . Given y′′ = (−4x4 + 2x2y2 + 2y4)/(x2y3), use this expression to show that P lies on a line y=mx and determine m exactly.
Zadanie 1f
Assume a<0 and b>a²/4. For P = (0, b) and : (i) find F″ and classify each stationary point; (ii) state whether these points lie above or below the x-axis.
Zadanie 11a
Find all points where the gradient is zero.
Zadanie 11b
Use second derivatives to classify the stationary points x=0,π/4,π/2. You may use .
Zadanie 11c
Sketch F on 0 ≤ x ≤ π/2. The endpoint maxima have height e+1 and second derivative −16(e+1); the middle minimum at π/4 has height e⁻¹−1 and second derivative 16(1+e⁻¹). Reflect these curvature sizes in the sketch.
Zadanie 11a
Find all points where the gradient is zero.
Zadanie 11b
Use second derivatives to classify the stationary points x=0,π/4,π/2. You may use .
Zadanie 11c
Sketch F on 0 ≤ x ≤ π/2. The endpoint maxima have height e+1 and second derivative −16(e+1); the middle minimum at π/4 has height e⁻¹−1 and second derivative 16(1+e⁻¹). Reflect these curvature sizes in the sketch.
Zadanie 1b
(i) State the coordinates of C₂’s two inflection points. (ii) Identify two features that distinguish C₁ from C₂.
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 2g
Let g(x) = (x − r)2[(x − a)2 + b2], b>0, with A=(a,g(a)) and R=(r,0). When a≠r the tangent at A has x-intercept T=((a+r)/2,0). Assume (a − r)2 > 2b2, so the quartic has two inflection points. (i) Show that their x-coordinates satisfy…
Zadanie 2h
Let g(x) = (x − r)2[(x − a)2 + b2], b>0, with A=(a,g(a)) and R=(r,0). When a≠r the tangent at A has x-intercept T=((a+r)/2,0). (i) Sketch the curve for a=r=−1 and b=2. (ii) For general a=r, state the coordinates of A and of the curve’s minimum point.