4.1 Related rates  (Page 4/7)

Find $\frac{dy}{dt}$ at $x=1$ and $y=x^2+3$ if $\frac{dx}{dt}=4.$

Find $\frac{dx}{dt}$ at $x=\mathrm{-2}$ and $y=2x^2+1$ if $\frac{dy}{dt}=\mathrm{-1}.$

Find $\frac{dz}{dt}$ at $\left(x,y\right)=\left(1,3\right)$ and $z^2=x^2+y^2$ if $\frac{dx}{dt}=4$ and $\frac{dy}{dt}=3.$

[T] If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, $\text{Ω})$ is given by the equation $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}.$ If $R_1$ is increasing at a rate of $0.5\text{Ω}\text{/}\text{min}$ and $R_2$ decreases at a rate of $1.1\text{Ω/min},$ at what rate does the total resistance change when $R_1=20\text{Ω}$ and $R_2=50\text{Ω}\text{/}\text{min}?$

A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

A 25-ft ladder is leaning against a wall. If we push the ladder toward the wall at a rate of 1 ft/sec, and the bottom of the ladder is initially $20\text{ft}$ away from the wall, how fast does the ladder move up the wall $5\text{sec}$ after we start pushing?