Q10 trial (1 Viewer)

[crammy90]

" for some a>0 two curves f(x) = a^x and g(x) = log(base a)x are drawn on the same axes so that they touch on y=x.
i) Write down expressions for their derivatives (Done)
ii) write down an equation involving natural logarithms whose solution is the x value at their points of contact
Answer:
since they touch on y=x
f'(x) = g'(x) = 1 at point of contact <---how do they know gradients are the same?
i.e. (a^x)lna = 1
a^x = 1/lna
so from ii
1/lna = (lnx)/(lna)
lnx = 1
x = e, y = e

i get everything except for how they knew the gradients would be 1 :S

 

[5647382910]

f(x) is the mirror image of g(x) about y = x. we know they touch the line y = x, ie. y = x is a tangent (gradient = 1) to both f(x) and g(x) at some point.

might save some confusion if you used a different pronumeral to 'x' to denote the point.
eg. f'(x₀) = g'(x₀) = 1 where x₀ is the x-value of the point of contact.

for a sketch go to: http://www.walterzorn.com/grapher/grapher_e.htm
and copy and paste 1.45^x; ln(x)/ln(1.45); x into the box, then go to plot graph

im lost... does a^x even touch loga (base a) x???
do both tese graphs even touch y = x???

 

[crammy90]

f(x) is the mirror image of g(x) about y = x. we know they touch the line y = x, ie. y = x is a tangent (gradient = 1) to both f(x) and g(x) at some point.

might save some confusion if you used a different pronumeral to 'x' to denote the point.
eg. f'(x₀) = g'(x₀) = 1 where x₀ is the x-value of the point of contact.

for a sketch go to: http://www.walterzorn.com/grapher/grapher_e.htm
and copy and paste 1.45^x; ln(x)/ln(1.45); x into the box, then go to plot graph

ok so i understand the graph from the copy and paste (thanks heaps)
but i dont understand how u knew they were opposites?
or in maths is g(x) denoted always to be the mirror of f(x)????
and even if u sub 1 into the equations how do u know that the graphs touch eachother at 1?
hmm actually y=x. the gradient of this is 1 as its at a 45 yeh?
so if they touch this line together, at this point there tangents will be both 1 yeh?
i was thinking about it like 1 graph could be y=x^2 and another be y=cosx and they wouldnt have the same gradients where they intersect at the line.
thats if they did intersect :S