only) Flashcards","topicId":"07681e27-ffd1-457b-97e1-6d9f7376045a","topicSlug":"differentiation-9","cards":[{"id":"b823e52d-2f67-4fe9-a712-c2e5eaf5bbff","cardType":"basic","front":"In which angle unit do the standard trigonometric derivatives apply directly?","back":"Radians.","points":1,"mcqOptions":null},{"id":"851f940a-3996-4551-8fba-9f94c1a1cbc7","cardType":"cloze","front":"For angles in radians, $\\dfrac{d}{dx}(\\sin x)=$ $\\text{[...]}$.","back":"For angles in radians, $\\dfrac{d}{dx}(\\sin x)=$ $\\boxed{\\cos x}$.","points":null,"mcqOptions":null},{"id":"48dff291-f76f-4d11-aba4-83779a6e54a4","cardType":"cloze","front":"For angles in radians, $\\dfrac{d}{dx}(\\cos x)=$ $\\text{[...]}$.","back":"For angles in radians, $\\dfrac{d}{dx}(\\cos x)=$ $\\boxed{-\\sin x}$.","points":null,"mcqOptions":null},{"id":"22ea8beb-ec1a-46e7-89c3-ad87e4f35f00","cardType":"basic","front":"What does $\\dfrac{f(x+h)-f(x)}{h}$ represent geometrically?","back":"The gradient of the secant through the two points.","points":1,"mcqOptions":null},{"id":"e2e2f9f3-996f-432c-a96b-6b90b5b3f203","cardType":"basic","front":"What does the secant approach as $h\\to0$?","back":"The tangent at $x$.","points":1,"mcqOptions":null},{"id":"f12064d5-d8fb-4838-83f7-a154e85479c8","cardType":"basic","front":"Why can $h=0$ not be substituted directly into the difference quotient?","back":"It would cause division by zero.","points":2,"mcqOptions":null},{"id":"24020003-f411-40f0-a666-a46a75a43464","cardType":"cloze","front":"The first-principles derivative is $\\text{[...]}$, provided the limit exists.","back":"The first-principles derivative is $\\boxed{f'(x)=\\displaystyle\\lim_{h\\to0}\\frac{f(x+h)-f(x)}{h}}$, provided the limit exists.","points":null,"mcqOptions":null},{"id":"ff23deaa-90af-42b9-b336-316cccdfbf88","cardType":"basic","front":"Why must radians be used in the standard trigonometric derivative proofs?","back":"The required small-angle limits have their standard values only in radians.","points":2,"mcqOptions":null},{"id":"caf789dc-98d2-4454-a36a-6adc1e024c27","cardType":"basic","front":"What would change in the derivatives of sine and cosine if angles were measured in degrees?","back":"An additional constant factor would appear.","points":1,"mcqOptions":null},{"id":"445c5aab-cefe-4bee-b182-8afac35edd79","cardType":"cloze","front":"For $h$ in radians, $\\text{[...]}$.","back":"For $h$ in radians, $\\boxed{\\displaystyle\\lim_{h\\to0}\\frac{\\sin h}{h}=1}$.","points":null,"mcqOptions":null},{"id":"94790a09-29a9-46b4-a6a3-076cbcbbe131","cardType":"cloze","front":"For $h$ in radians, $\\text{[...]}$.","back":"For $h$ in radians, $\\boxed{\\displaystyle\\lim_{h\\to0}\\frac{\\cos h-1}{h}=0}$.","points":null,"mcqOptions":null},{"id":"809e9554-d5f6-4d89-80ff-c181f6190d3b","cardType":"basic","front":"Why does $\\dfrac{\\sin h}{h}\\to1$ as $h\\to0$ in radians?","back":"For small $h$, $\\sin h$ approaches $h$.","points":2,"mcqOptions":null},{"id":"73456830-8f63-41b4-a08e-5c7a52d730f6","cardType":"basic","front":"Which identity is used to derive the second small-angle limit?","back":"$1-\\cos h=2\\sin^2\\left(\\dfrac{h}{2}\\right)$.","points":1,"mcqOptions":null},{"id":"bf30229f-c076-4c1f-8e9b-5d1895731521","cardType":"basic","front":"Rewrite $\\dfrac{\\cos h-1}{h}$ so the first small-angle limit appears.","back":"$-\\sin\\left(\\dfrac{h}{2}\\right)\\dfrac{\\sin\\left(\\frac{h}{2}\\right)}{\\frac{h}{2}}$.","points":2,"mcqOptions":null},{"id":"c4515179-9a88-4687-b045-7c25e9570a36","cardType":"basic","front":"Why does the rewritten expression for $\\dfrac{\\cos h-1}{h}$ tend to zero?","back":"Its factors tend to $0$ and $1$, so their product tends to $0$.","points":2,"mcqOptions":null},{"id":"5cbacba0-261e-4e2d-9be9-686754e06101","cardType":"cloze","front":"The sine addition formula gives $\\sin(x+h)=$ $\\text{[...]}$.","back":"The sine addition formula gives $\\sin(x+h)=$ $\\boxed{\\sin x\\cos

3.5 Differentiation (A-level only) Flashcards

3.5 Differentiation (A-level only) Flashcards