:"35043466-d6e9-4e8c-9da8-d266578613e9","topicSlug":"differentiation-7","cards":[{"id":"af0a92c1-022e-4a12-9621-2a083778fff7","cardType":"basic","front":"What does evaluating $f'(a)$ give?","back":"The curve's gradient at $x=a$.","points":1,"mcqOptions":null},{"id":"53763437-4830-4a2b-9be4-d13af5dd4f36","cardType":"cloze","front":"The power rule is $\\frac{d}{dx}(x^n)=$ $\\text{[...]}$.","back":"The power rule is $\\frac{d}{dx}(x^n)=$ $\\boxed{nx^{n-1}}$.","points":null,"mcqOptions":null},{"id":"7bdf0c6d-a9f0-41d2-a42c-a015da698a22","cardType":"cloze","front":"A constant differentiates to [ ].","back":"A constant differentiates to zero.","points":null,"mcqOptions":null},{"id":"ca02f927-a78f-492e-84b8-143a867d77fc","cardType":"basic","front":"Differentiate $y=4x^3-5x^2+7x-9$.","back":"$$\\frac{dy}{dx}=\\mathbf{12x^2-10x+7}$.","points":1,"mcqOptions":null},{"id":"c071449d-c4f4-447d-9659-72e78b0f90e4","cardType":"basic","front":"Why must you differentiate before substituting a given value of $x$?","back":"Early substitution produces a single number, losing gradient information.","points":2,"mcqOptions":null},{"id":"0fa45b2c-2ea2-4af5-b3d8-36e3f5bfb5d8","cardType":"basic","front":"Find the gradient of $y=2x^3-3x^2+4$ at $x=2$.","back":"The gradient is $12$.","points":2,"mcqOptions":null},{"id":"6acf9574-ae96-4873-a309-9d65f96f1033","cardType":"cloze","front":"At $x=a$, the tangent gradient is $\\text{[...]}$.","back":"At $x=a$, the tangent gradient is $\\boxed{m_{\\text{tangent}}=f'(a)}$.","points":null,"mcqOptions":null},{"id":"93e8906a-e622-40a3-ad0a-b352250189c7","cardType":"basic","front":"How is the point of contact found when its $x$-coordinate is known?","back":"Substitute the $x$-coordinate into the original curve equation.","points":1,"mcqOptions":null},{"id":"b763389b-1b1c-4cba-9bab-126745e30603","cardType":"cloze","front":"The point-gradient equation of a straight line is $\\text{[...]}$.","back":"The point-gradient equation of a straight line is $\\boxed{y-y_1=m(x-x_1)}$.","points":null,"mcqOptions":null},{"id":"75927dd3-86e0-4ee5-87c9-70aab232b275","cardType":"basic","front":"Find the tangent to $y=x^2+3x-1$ at $x=2$.","back":"$y=7x-5$.","points":3,"mcqOptions":null},{"id":"0fd1d47e-cf8a-4547-bd0b-8eb32ff40fc8","cardType":"cloze","front":"For two non-vertical perpendicular lines, $\\text{[...]}$.","back":"For two non-vertical perpendicular lines, $\\boxed{m_1m_2=-1}$.","points":null,"mcqOptions":null},{"id":"f6f82ae4-c2d3-4a51-9f73-62b1bede8920","cardType":"basic","front":"If a tangent has non-zero gradient $m$, what is the normal gradient?","back":"$-\\frac{1}{m}$, the negative reciprocal.","points":1,"mcqOptions":null},{"id":"6e41ce9f-74dd-4dbe-8247-a1e8d8156a9c","cardType":"cloze","front":"If the tangent at $x=a$ is horizontal, the normal is the vertical line $\\text{[...]}$.","back":"If the tangent at $x=a$ is horizontal, the normal is the vertical line $\\boxed{x=a}$.","points":null,"mcqOptions":null},{"id":"a4de5c98-7ce9-480c-bda0-957ceb2e1a38","cardType":"basic","front":"What equation is solved to find the $x$-coordinates of stationary points?","back":"$\\frac{dy}{dx}=0$.","points":1,"mcqOptions":null},{"id":"e33f5fb4-54ec-4079-9be1-03eadb41a289","cardType":"basic","front":"After finding a stationary point's $x$-coordinate, how is its $y$-coordinate found?","back":"Substitute $x$ into the original equation.","points":1,"mcqOptions":null},{"id":"94f73417-b212-4264-bc3a-6438dca59bc6","cardType":"basic","front":"How is $\\frac{d^2y}{dx^2}$ obtained?","back":"Differentiate $\\frac{dy}{dx}$ again.","points":1,"mcqOptions":null},{"id":"3c05e418-fceb-4138-a556-cce7afe832e0","cardType":"basic","front":"What does $f''(a)>0$ show at a stationary point?","back":"It is a local minimum.","points":1,"mcqOptions":null},{"id":"5b8944cd-fbf7-46d0-

1.6.6 Differentiation Flashcards

1.6.6 Differentiation Flashcards