egration Flashcards","topicId":"72ab2f83-019a-4b6a-ae63-e56f24c0361c","topicSlug":"constant-of-integration","cards":[{"id":"e5f7a05a-e4a6-43e5-a3da-1f0e48f1601d","cardType":"basic","front":"Why must an indefinite integral include $+C$?","back":"Because constants disappear when differentiated, so all possible antiderivatives differ by a constant.","points":2,"mcqOptions":null},{"id":"1685d839-7184-4f45-b0ad-bd35fd142e07","cardType":"cloze","front":"If $F'(x)=f(x)$, then $\\int f(x)\\,dx=$ $\\text{[...]}$.","back":"If $F'(x)=f(x)$, then $\\int f(x)\\,dx=$ $\\boxed{F(x)+C}$.","points":null,"mcqOptions":null},{"id":"16b061ab-03c4-4b66-98d5-09b9e02ab28c","cardType":"basic","front":"Why do $y=x^3+2$ and $y=x^3-9$ have the same derivative?","back":"Both constants differentiate to zero.","points":2,"mcqOptions":null},{"id":"8c0f6370-f242-45e4-b206-1934c2e998fb","cardType":"basic","front":"What happens to the curve $y=x^2+C$ when $C$ increases?","back":"The whole curve moves vertically upwards without changing shape.","points":1,"mcqOptions":null},{"id":"ce94b84a-eb29-4a76-99a4-719d276c2cc8","cardType":"basic","front":"What does knowing a curve's derivative fail to determine?","back":"Its vertical position.","points":1,"mcqOptions":null},{"id":"cbc36318-1d4e-4b8b-81f5-23948ea50804","cardType":"cloze","front":"All curves in $y=x^2+C$ have the same [ ] at each value of $x$.","back":"All curves in $y=x^2+C$ have the same gradient at each value of $x$.","points":null,"mcqOptions":null},{"id":"85fdae17-cc8f-4990-8e83-392899530ec2","cardType":"basic","front":"What does an initial condition do to a family of antiderivatives?","back":"It selects one particular function from the family.","points":1,"mcqOptions":null},{"id":"e65ea085-3a4c-410b-9aff-01e3e0c3470c","cardType":"basic","front":"What point on $y=f(x)$ is represented by $f(3)=10$?","back":"$(3,10)$","points":1,"mcqOptions":null},{"id":"998189ed-afac-414a-b64d-01d20d266bb7","cardType":"basic","front":"Where should a known point $(a,b)$ be substituted to determine $C$?","back":"Into the integrated equation for $y$.","points":1,"mcqOptions":null},{"id":"b86b7cbb-05e4-4258-b605-49b72cfe790a","cardType":"cloze","front":"A point $(a,b)$ on a curve means that [ ] satisfy its equation.","back":"A point $(a,b)$ on a curve means that $x=a$ and $y=b$ satisfy its equation.","points":null,"mcqOptions":null},{"id":"b49664da-f2e6-4d1b-8583-aeb47eed32d2","cardType":"basic","front":"Find the general function if $\\frac{dy}{dx}=6x^2-4x+3$.","back":"$y=2x^3-2x^2+3x+C$","points":2,"mcqOptions":null},{"id":"ca3f50e1-2605-4d1b-a954-832c8dc7b72a","cardType":"basic","front":"Given $\\frac{dy}{dx}=2x-5$, find the general equation of the curve.","back":"$y=x^2-5x+C$","points":2,"mcqOptions":null},{"id":"9e011c17-0a4b-41ae-947d-f41bca0302a4","cardType":"basic","front":"Given $\\frac{dy}{dx}=2x-5$ and $y=7$ when $x=3$, find $C$.","back":"$C=13$","points":2,"mcqOptions":null},{"id":"bdc98070-6388-4d22-9af7-7eb9ecea684f","cardType":"basic","front":"Given $\\frac{dy}{dx}=2x-5$ and $y=7$ when $x=3$, find the curve's equation.","back":"$y=x^2-5x+13$","points":2,"mcqOptions":null},{"id":"f00d5494-3f38-4cd3-9cbd-8f8ed9aedd78","cardType":"basic","front":"A curve has $\\frac{dy}{dx}=4x^3-6x$. What is its general equation?","back":"$y=x^4-3x^2+C$","points":2,"mcqOptions":null},{"id":"a278569e-d6ac-4dc0-9741-9a8a34b2e355","cardType":"cloze","front":"For $y=x^4-3x^2+C$ passing through $(2,7)$, the constant is $\\text{[...]}$.","back":"For $y=x^4-3x^2+C$ passing through $(2,7)$, the constant is $\\boxed{C=3}$.","points":null,"mcqOptions":null},{"id":"ee15acae-2a1d-4a46-86dc-7bbaacdf1e84","cardType":"basi

1.9.21 Constant of integration Flashcards

1.9.21 Constant of integration Flashcards