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MIT OpenCourseWare18.01 Single Variable CalculusFall 2006For information about citing these materials or our Terms of Use, visit:.
Lecture 618.01 Fall 2006Lecture 6: Exponential and Log, LogarithmicDierentiation, Hyperbolic FunctionsffTaking the derivatives of exponentials and logarithmsBackgroundWe always assume the base, a, is greater than 1.a0= 1;a1= a;a2= a · a;. . .ax1+x2=ax1ax2(ax1)x2=ax1x2pa=√qap(where p and q are integers)qTo define arfor real numbers r, fill in by continuity.Today’s main task: finddaxdxWe can writedax+Δxaxax=limdxΔx→0ΔxWe can factor out the ax:limax+Δxax=lim axaΔx− 1= axlimaΔx− 1ΔxΔxΔx→0Δx→0ΔxΔx→0Let’s callM(a) ≡ limaΔx1ΔxWe don’t yet know what M(a) is, but we can saydxdax= M(a)axHere are two ways to describe M(a):1. Analytically M(a) =dxdaxat x = 0.Indeed, M(a) =a0+Δxa0dxx=0Δx→0Δx−=dxalim1Δx→0
Lecture 618.01 Fall 2006axM(a)(slope of axat x=0)Figure 1:Geometric definition of M(a)2. Geometrically, M(a) is the slope of the graph y = axat x = 0.The trick to figuring out what M(a) is is to beg the question and definee as the number suchthat M(e) = 1. Now can we be sure there is such a number e? First notice that as the base aincreases, the graph axgets steeper. Next, we will estimate the slope M(a) for a = 2 and a = 4geometrically. Look at the graph of 2xin Fig.

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Term
Fall
Professor
BROWN
Tags
lim, Natural logarithm, Logarithm
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