https://teddyrookbook.com/math/calculus-1/Recent content in Calculus 1: Limits and Derivatives on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/calculus-1/limit-idea/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/limit-idea/lim f(x) = L when f(x) gets close to L as x gets close to a, from both sides. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/one-sided-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/one-sided-limits/The two-sided limit exists only when the left and right limits agree. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/infinite-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/infinite-limits/f(x) can grow without bound near a point. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/limit-laws/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/limit-laws/Limits of sums, products and quotients follow from the limits of the parts. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/limits-by-factoring/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/limits-by-factoring/Rewrite 0/0 forms to remove the troublesome factor. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/limits-by-rationalizing/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/limits-by-rationalizing/Multiply by a conjugate to resolve 0/0 forms with roots. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/squeeze-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/squeeze-theorem/Trap a function between two with the same limit. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/trig-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/trig-limits/sin(x)/x → 1 and (1 − cos x)/x → 0 as x → 0. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/continuity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/continuity/f is continuous at a when the limit equals f(a). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/intermediate-value-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/intermediate-value-theorem/A continuous function that changes sign must hit 0 in between. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/limits-at-infinity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/limits-at-infinity/Compare the growth of the highest powers. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/epsilon-delta/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/epsilon-delta/For every ε > 0 there is a δ > 0 that keeps f(x) within ε of L. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/instantaneous-rate/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/instantaneous-rate/Shrink the interval of an average rate to get a rate at an instant. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/derivative-at-a-point/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/derivative-at-a-point/f′(a) is the limit of the difference quotient: the slope of the tangent line. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/tangent-lines/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/tangent-lines/Use f′(a) as the slope of the line y = f(a) + f′(a)(x − a). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/derivative-function/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/derivative-function/f′ gives the slope of f at every x. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/differentiability/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/differentiability/Corners, cusps, vertical tangents and breaks. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/derivative-notation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/derivative-notation/dy/dx is a rate with units of y per unit of x. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/power-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/power-rule/d/dx xⁿ = nxⁿ⁻¹, and derivatives split over sums. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/product-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/product-rule/(fg)′ = f′g + fg′. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/quotient-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/quotient-rule/(f/g)′ = (f′g − fg′)/g². Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/trig-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/trig-derivatives/d/dx sin x = cos x and d/dx cos x = −sin x. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/higher-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/higher-derivatives/Differentiate again to get f″, f‴ and beyond. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/chain-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/chain-rule/(f(g(x)))′ = f′(g(x)) · g′(x). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/chain-rule-layers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/chain-rule-layers/Peel a composition from the outside in. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/exp-log-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/exp-log-derivatives/d/dx eˣ = eˣ and d/dx ln x = 1/x. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/implicit-differentiation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/implicit-differentiation/Differentiate both sides, treating y as a function of x. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/inverse-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/inverse-derivatives/d/dx sin⁻¹ x = 1/√(1 − x²). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/log-differentiation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/log-differentiation/Take ln of both sides to handle powers like xˣ. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/motion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/motion/Velocity is the derivative of position; acceleration is the derivative of velocity. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/related-rates/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/related-rates/Differentiate an equation linking quantities to connect their rates. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/linear-approximation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/linear-approximation/Near a, f(x) ≈ f(a) + f′(a)(x − a). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/critical-points-extrema/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/critical-points-extrema/Check critical points and endpoints on a closed interval. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/mean-value-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/mean-value-theorem/Somewhere the instantaneous rate equals the average rate. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/first-derivative-test/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/first-derivative-test/The sign of f′ tells where f rises or falls. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/concavity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/concavity/The sign of f″ tells which way the graph bends. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/curve-sketching/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/curve-sketching/Combine limits, f′ and f″ to sketch a graph. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/optimization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/optimization/Model a quantity, then find its largest or smallest value. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/lhopital/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/lhopital/For 0/0 or ∞/∞, compare the derivatives of the numerator and denominator. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/newtons-method/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/newtons-method/Use tangent lines to approach a root. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/antiderivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/antiderivatives/Run differentiation backward, and add the constant C. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/initial-value-antiderivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/initial-value-antiderivatives/Use a known value to find C. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/area-approximations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/area-approximations/Add up rectangle areas to estimate the area under a curve. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/riemann-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/riemann-sums/A sum of f(xᵢ)Δx approximates the area; more rectangles do better. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/definite-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/definite-integral/The limit of Riemann sums gives net signed area. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/integral-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/integral-properties/Split intervals, reverse limits and compare integrals. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/ftc-accumulation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/ftc-accumulation/d/dx ∫ₐˣ f(t) dt = f(x). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/ftc-evaluate/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/ftc-evaluate/∫ₐᵇ f(x) dx = F(b) − F(a). Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/net-change/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/net-change/The integral of a rate gives net change; total distance needs |v|. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/average-value/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/average-value/f_avg = (1/(b − a)) ∫ₐᵇ f(x) dx. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/u-substitution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/u-substitution/Reverse the chain rule by choosing u = inside function. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/substitution-definite/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/substitution-definite/Change the limits to match u. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/exp-log-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/exp-log-integrals/∫ 1/x dx = ln|x| + C and ∫ eˣ dx = eˣ + C. Calculus 1: Limits and Derivatives · Math Ladderhttps://teddyrookbook.com/math/calculus-1/inverse-trig-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-1/inverse-trig-integrals/∫ 1/(1 + x²) dx = tan⁻¹ x + C. Calculus 1: Limits and Derivatives · Math Ladder