Maths : Exponential / Log function?

Question

1. Given that a curve has the equation y=10(x+2)ˆx, find dy/dx and hence find the coordinates of the stationary points. Find the coordinates of the points at which the curve crosses the x- and y-axes.

2. An object is heated in an oven until it reaches a temperature of X degrees Celsius. It is then allowed to cool. Its temperature, θ degrees Celsius, wehn it has been cooling for time t minutes, is given by the equation θ = 18+62eˆ(t/8)

Find
a. the value of X
b. the value of θ when t = 16
c. the value of t when θ = 48
d. the rate at which θ is decreasing when t = 16
State the value which θ approaches as t becomes very large.

Answer 1

y=10*(x+2)^x = 10* e^x*(ln(x+2)) so

dy/dx= 10 e^xln(x+2) *(ln(x+2)+x/(x+2))

ln(x+2)+x/(x+2)=0 x>-2
ln(x+2)-2/(x+2) +1=0 If we put x+2=z
lnz-2/z+1=0

f(z)=lnz-2/z+1
limf z=>0+=-infinity
limf z=>+infinity = +infinity
f´(z) = 1/z+4/z^2 >0
so fz is always increasing and has only one root
f(1)<0 and f(2)>0
so the root is between 1 and 2 for z and between -1 and0 forx

An exponential is always>0 so it never crosses the x axis
forx=0 y=10 (y cross)
2)@=18+62e-^(t/8)

X= 18+62= 80 oC
@(16) = 18 +62 e^-2=26.39 oC
48= 18 +62 e^-(t/8) e^(-t/8) = 30/62 and t = -8*ln(30/62)=5.81 minutes
d@/dt=-62/8e^(-t/8) = -1.049 oC/min at t=16

d)18 oC

Answer 2

ok = 5 Tangent Line at x = 2 is: y = 3x - 6 commonplace Line at x = 2 is: y = -(a million/3) x + 2/3 (oops, had in simple terms the tangent line earlier) Do i'm getting a cookie :-) And now the artwork .... For ok, given the line crosses the x-axis at x=2, the equation turns into: 0 = ln (3(2) - ok) -> 0 = ln ( 6-ok ) taking e^x on the two sides we get e^0 = e^ (ln (6-ok ) ) e to the 0 is a million anfd e ^ ln (u) = u.s. e ^ ln (6-ok) is 6-ok this provides us a million = 6 - ok or ok = 5 (ding ding ding) to locate the line commonplace to the curve, first locate the line tangent to the curve,. The slope of this line is comparable to the fee of substitute of the curve on the element of intersection whci may be got here across via differentiation. d/dx ln(u) = a million/u ( d/du (u) ) d/dx ( ln (3x-ok)) = a million/(3x-ok) d/dx (3x-ok) = a million/(3x-ok) 3 = 3/(3x-ok) at x=2 and ok=5, the slope is 3 With a slope of three and crossing the x-axis at x=2, the equation of the tangent line is: y = 3x - 6 the traditional line is perpendicular to the tangent line so that's equation is: y = -(a million/3) x + 2/3 (wow, takes a protracted time to kind this with in simple terms ascii text textile, desire that's sparkling) - Bert