":"What does $f'(x)$ tell you about a curve?","back":"Its gradient at each value of $x$.","points":1,"mcqOptions":null},{"id":"a9599421-5097-4805-afbe-dc38ac32f65c","cardType":"basic","front":"What does $f''(x)$ measure?","back":"The rate of change of the gradient with respect to $x$.","points":1,"mcqOptions":null},{"id":"ad5740bf-1638-4b70-bf17-54dd6affbd0e","cardType":"basic","front":"What happens to the gradient where $f''(x)>0$?","back":"The gradient is increasing.","points":1,"mcqOptions":null},{"id":"533a81f7-9420-4f4c-bf64-148bd9bd22f4","cardType":"basic","front":"What happens to the gradient where $f''(x)<0$?","back":"The gradient is decreasing.","points":1,"mcqOptions":null},{"id":"bf07abe5-d571-49fb-b812-074b27857714","cardType":"cloze","front":"For $y=f(x)$, the gradient is $\\text{[...]}$, while the rate of change of gradient is $f''(x)=\\frac{d^2y}{dx^2}$.","back":"For $y=f(x)$, the gradient is $\\boxed{f'(x)=\\frac{dy}{dx}}$, while the rate of change of gradient is $f''(x)=\\frac{d^2y}{dx^2}$.","points":null,"mcqOptions":null},{"id":"8699de99-bee6-41f7-acdf-80f84c75db25","cardType":"cloze","front":"For $y=f(x)$, the gradient is $f'(x)=\\frac{dy}{dx}$, while the rate of change of gradient is $\\text{[...]}$.","back":"For $y=f(x)$, the gradient is $f'(x)=\\frac{dy}{dx}$, while the rate of change of gradient is $\\boxed{f''(x)=\\frac{d^2y}{dx^2}}$.","points":null,"mcqOptions":null},{"id":"de824e1e-49cb-4236-8ce5-a85f422b000d","cardType":"basic","front":"What does $f''(a)=0$ indicate about the gradient at $x=a$?","back":"It is momentarily not changing, but further investigation is needed.","points":1,"mcqOptions":null},{"id":"5452d5a3-86b1-4b3b-afca-a0a1d1685ee0","cardType":"cloze","front":"A curve is [ ] where $f''(x)>0$ and concave where $f''(x)<0$.","back":"A curve is convex where $f''(x)>0$ and concave where $f''(x)<0$.","points":null,"mcqOptions":null},{"id":"c65c2f96-2274-49cf-81b4-bbf5a5d7a52d","cardType":"cloze","front":"A curve is convex where $f''(x)>0$ and [ ] where $f''(x)<0$.","back":"A curve is convex where $f''(x)>0$ and concave where $f''(x)<0$.","points":null,"mcqOptions":null},{"id":"f5d19ed6-17f5-49cd-a9fb-53fe69200aec","cardType":"basic","front":"Can a function be decreasing where $f''(x)>0$? Explain.","back":"Yes - its gradient can be negative but increasing.","points":2,"mcqOptions":null},{"id":"a9cde120-5916-444b-9399-5a1928393c0f","cardType":"basic","front":"What change confirms a point of inflection?","back":"$$f''(x)$ changes sign across the point.","points":1,"mcqOptions":null},{"id":"74618dde-030a-45af-a40f-bbfd5af54d53","cardType":"basic","front":"Why does $f''(a)=0$ not prove that $x=a$ is a point of inflection?","back":"The second derivative may not change sign across $a$.","points":2,"mcqOptions":null},{"id":"d221b443-fbf1-4544-90e0-840dc24d51b1","cardType":"basic","front":"Why is the origin not a point of inflection for $f(x)=x^4$?","back":"$$f''(x)=12x^2$ is positive on both sides of zero.","points":2,"mcqOptions":null},{"id":"266b47d1-681e-4f39-9477-21a49120cdfa","cardType":"basic","front":"How do you identify a stationary point of inflection at $x=a$?","back":"Concavity changes and $f'(a)=0$.","points":2,"mcqOptions":null},{"id":"2be137cd-c698-4f58-a7d3-68364ecdc8c3","cardType":"basic","front":"How do you identify a non-stationary point of inflection at $x=a$?","back":"Concavity changes and $f'(a)\\neq0$.","points":2,"mcqOptions":null},{"id":"a433a46b-c1fe-490e-a9b2-052d22ab3aa3","cardType":"basic","front":"Where else can a change of concavity occur besides where $f''(x)=0$?","back":"Where $f''(x)$ is undefined.","points":1,"mcqOptions":null},{"id":"5133e325-1

3.6.2 Differentiation (A-level only) Flashcards

3.6.2 Differentiation (A-level only) Flashcards