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Maths question(The exponential function in derivative)

發問:

For what values of a and b are y = a b^x and y = 4 + x tangent at x = 0. a = ? b = ?

最佳解答:

x=0, y=4+0=4, y= a b^x, at x=0, y=4, 4= a y=4b^x, ln(y)=ln(4b^x)= ln(4) + ln(b^x)= ln(4) +xln(b) d[ln(y)]/dx=ln(b) (1/y)(dy/dx)=ln(b) dy/dx=yln(b)=1 at x=0 (same slope as y=4+x at x=0) 4ln(b)=1, b=1.284

其他解答:

The equation of tangent of y=a*b^x at x=0 is y=y(0)+y'(0)x=a+a lnb x since given that y=4+x thus a=4 and a lnb=1, i.e. a=4, b=e^(1/4)