Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Differentiation, Exercise 4: EXERCISE 4C

Find the gradient of the function $y=1-{\mathrm{e}}^{2-x}.$

Find the equation of the normal to the curve $y=1-{\mathrm{e}}^{2-x}$ at the point where $y=0.$

The mass, $m$ grams, of a radioactive substance remaining $t$ years after a given time, is given by the formula $m=300{\mathrm{e}}^{-0.00012t}.$ Find the rate at which the mass is decreasing when $t=2000.$

The equation of a curve is $y=x^2{e}^{-x}.$

Show that the equation of the normal to the curve at the point where $x=1$ is $e^{2}x+ey=1+e^2$.

Find the exact value of the $x$ -coordinates of the points on the curve $y=x^2{e}^{-2x}$ at which $\frac{d^{2}y}{ d{x}^2}=0.$

Find the coordinates of the stationary point on the curve $y=\frac{e^{2x-1}}{x}.$

By writing $2$ as $e^{\ln 2}$ prove that $\frac{d}{dx}\left(2^x\right)=2^x\ln 2$

The equation of a curve is $y=x\left(3^x\right).$
Find the exact value of the gradient of the tangent to the curve at the point where $x=1.$