Using our knowledge of the natural logarithm function and its inverse relationship with the exponential function, we can apply our newly acquired tools in solving equations and calculus to a wide range of applications. We can now find algebraic solutions for exponential problems where previously we required technology, as well as being able to apply calculus to general exponential functions.
Keep the following rules in mind as we look at some examples.
Function | Derivative |
---|---|
$e^{f\left(x\right)}$ef(x) | $f\ '\left(x\right)e^{f\left(x\right)}$f ′(x)ef(x) |
$\ln\left(f\left(x\right)\right)$ln(f(x)) | $\frac{f\ '\left(x\right)}{f\left(x\right)}$f ′(x)f(x) |
Function $f\left(x\right)$f(x) | Integral $\int f\left(x\right)dx$∫f(x)dx |
---|---|
$e^{ax+b}$eax+b | $\frac{1}{a}e^{ax+b}+C$1aeax+b+C |
$\frac{1}{ax+b}$1ax+b | $\frac{1}{a}\ln\left(ax+b\right)+C$1aln(ax+b)+C, for $ax+b>0$ax+b>0 |
The gradient of a curve at any point is given by $f\ '\left(x\right)=\frac{6}{x-3}$f ′(x)=6x−3, and the curve passes through the point $\left(4,2\right)$(4,2) and is defined for $x>3$x>3.
Find an expression for the function $f(x)$f(x).
Think: Use integration to find the family of curves with the gradient function $f\ '\left(x\right)=\frac{6}{x-3}$f ′(x)=6x−3 and then use the given point to solve for the constant of integration, $C$C.
Do:
$f\left(x\right)$f(x) | $=$= | $\int\frac{6}{x-3}dx$∫6x−3dx |
$=$= | $6\int\frac{1}{x-3}dx$6∫1x−3dx | |
$=$= | $6\ln\left(x-3\right)+C$6ln(x−3)+C | |
Using the given point $\left(4,2\right)$(4,2), substitute into the function to solve for $C$C.
$2$2 | $=$= | $6\ln\left(4-3\right)+C$6ln(4−3)+C |
$2$2 | $=$= | $6\ln1+C$6ln1+C |
$\therefore$∴ $C$C | $=$= | $2$2 |
The function is given by $f\left(x\right)=6\ln\left(x-3\right)+2$f(x)=6ln(x−3)+2.
A plane takes off from an airport at sea level and its altitude $h$h in metres, $t$t minutes after taking off, is given by $h=600\ln\left(t+1\right)$h=600ln(t+1).
Exactly $t$t minutes after taking off, at what rate is the plane ascending?
Hence, what is the rate of ascent at exactly $4$4 minutes after take off?
How would you describe the ascent of the plane?
Ascending, at an increasing rate.
Ascending, but at a decreasing rate.
A circus tent is $7$7 m high and has a radius of $6$6 m. The equation to describe the curved roof of the tent is $y=\frac{7}{x+1}$y=7x+1, as shown in the graph.
Calculate the cross-sectional area of the tent. Leave your answer in exact form.