Rates of change questions with answers

Questions

1. Question 1Multiply by the rate given

   A point moves along the curve $y = x^{3} - 2x^{2} + 1$ so that its $x$-coordinate is increasing at a rate of $3$ units per second. Find the rate of change of its $y$-coordinate when $x = -1$.

   Show the worked solution

   1. Step 1: Differentiate y with respect to x$\dfrac{dy}{dx} = 3x^{2} - 2(2)x$$= 3x^{2} - 4x$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = 3(-1)^{2} - 4(-1)$$= 7$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dy}{dt} = \dfrac{dy}{dx} \times \dfrac{dx}{dt}$
   4. Step 4: Put the values in$\dfrac{dy}{dt} = 7 \times 3$$= 21$ units/s
2. Question 2Multiply by the rate given

   A point moves along the curve $y = -x^{3} + 2x^{2} + 5$ so that its $x$-coordinate is increasing at a rate of $0.2$ units per second. Find the rate of change of its $y$-coordinate when $x = 2$.

   Show the worked solution

   1. Step 1: Differentiate y with respect to x$\dfrac{dy}{dx} = -3x^{2} + 2(2)x$$= -3x^{2} + 4x$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = -3(2)^{2} + 4(2)$$= -4$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dy}{dt} = \dfrac{dy}{dx} \times \dfrac{dx}{dt}$
   4. Step 4: Put the values in$\dfrac{dy}{dt} = -4 \times 0.2$$= -\dfrac{4}{5}$ units/s
3. Question 3Divide the rate given

   A point moves along the curve $y = 2x^{2} + x - 5$ so that its $y$-coordinate is increasing at a rate of $3$ units per second. Find the rate of change of its $x$-coordinate when $x = 1$.

   Show the worked solution

   1. Step 1: Differentiate y with respect to x$\dfrac{dy}{dx} = 2(2)x + 1$$= 4x + 1$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = 4(1) + 1$$= 5$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dx}{dt} = \dfrac{dy}{dt} \div \dfrac{dy}{dx}$
   4. Step 4: Put the values in$\dfrac{dx}{dt} = 3 \div 5$$= \dfrac{3}{5}$ units/s
4. Question 4Divide the rate given

   A point moves along the curve $y = x^{3} - 4x$ so that its $y$-coordinate is increasing at a rate of $3$ units per second. Find the rate of change of its $x$-coordinate when $x = 2$.

   Show the worked solution

   1. Step 1: Differentiate y with respect to x$\dfrac{dy}{dx} = 3x^{2} - 4(1)$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = 3(2)^{2} - 4$$= 8$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dx}{dt} = \dfrac{dy}{dt} \div \dfrac{dy}{dx}$
   4. Step 4: Put the values in$\dfrac{dx}{dt} = 3 \div 8$$= \dfrac{3}{8}$ units/s
5. Question 5Circles, spheres and cylinders

   The radius of a circle is increasing at a rate of $0.2$ cm/s. Find the rate of increase of its area when the radius is $3$ cm, leaving your answer in terms of $\pi$.

   Show the worked solution

   1. Step 1: Write the formula for A in terms of r, and differentiate it$A = \pi r^{2}$$\dfrac{dA}{dr} = 2\pi r$
   2. Step 2: Put in the given value of r$\dfrac{dA}{dr} = 2\pi (3)$$= 6\pi$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt}$
   4. Step 4: Put the values in$\dfrac{dA}{dt} = 6\pi \times 0.2$$= \dfrac{6\pi}{5}$ cm²/s
6. Question 6Circles, spheres and cylinders

   The radius of a circle is increasing at a rate of $4$ cm/s. Find the rate of increase of its area when the radius is $10$ cm, leaving your answer in terms of $\pi$.

   Show the worked solution

   1. Step 1: Write the formula for A in terms of r, and differentiate it$A = \pi r^{2}$$\dfrac{dA}{dr} = 2\pi r$
   2. Step 2: Put in the given value of r$\dfrac{dA}{dr} = 2\pi (10)$$= 20\pi$
   3. Step 3: Connected rates of change: link the rate you want to the rate you are given$\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt}$
   4. Step 4: Put the values in$\dfrac{dA}{dt} = 20\pi \times 4$$= 80\pi$ cm²/s