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Wize High School Grade 12 Pre-Calculus Textbook > Rates of Change

Rates of Change of Polynomials

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Rates of Change of Polynomials


The rate of change of a polynomial P(x)=axn+bxn−1+...+cP(x)=ax^n+bx^{n-1}+...+c, where a,b∈Ra,b\in\mathbb{R} and n∈Wn\in\mathbb{W}, can be solved used both algebraic and graphical methods.

Example

Let P(x)=2x6−3x3+x2−1P(x)=2x^6-3x^3+x^2-1.

The instantaneous rate of change of P(x)P(x) at x=1x=1 is:

Graphical Solution

Graph P(x)=2x6−3x3+x2−1P(x)=2x^6-3x^3+x^2-1 using a graphing calculator.


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Sketch the tangent line by pressing 2nd + PRGM + 5: Tangent + ENTER\boxed{2^{\text{nd}}}~+~\boxed{\text{PRGM}}~+~\boxed{\text{5: Tangent}}~+~\boxed{\text{ENTER}}


The equation of the tangent line is y=5x−6y=5x-6.

The rate of change is the slope of the tangent line. Therefore, the instantaneous rate of change of P(x)P(x) at x=1x=1 is 55.


Algebraic Solution

Using the difference quotient P(a+h)−P(a)h\dfrac{P(a+h)-P(a)}{h} and choosing a very small value for hh and a=1a=1, we can estimate for the instantaneous values:

P(a+h)−P(a)h=P(1.01)−P(1)0.01=(2(1.01)6−3(1.01)3+(1.01)2−1)−(2(1)6−3(1)3+(1)2−1)0.01≈5.224\begin{array}{rcl} \dfrac{P(a+h)-P(a)}{h}&=&\dfrac{P(1.01)-P(1)}{0.01}\\\\ &=&\dfrac{(2(1.01)^6-3(1.01)^3+(1.01)^2-1)-(2(1)^6-3(1)^3+(1)^2-1)}{0.01}\\\\ &\approx&5.224 \end{array}
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Example: Rates of Change of Polynomials

Let f(x)=−4x3+x2−1f(x)=-4x^3+x^2-1.
  1. Determine the average rate of change for each interval:
  2. 0≤x≤40\leq{}x\leq{}4
  3. 0≤x≤20\leq{}x\leq{}2
  4. 2≤x≤42\leq{}x\leq{}4
  5. Estimate the instantaneous rate of change when x=2x=2.
1.

i. msec=(4x3+x2−1)∣x=4−(4x3+x2−1)∣x=04−0=(4(4)3+(4)2−1)−(−1)4=68\begin{array}{rcl} m_{sec}&=&\dfrac{(4x^3+x^2-1)|_{x=4}-(4x^3+x^2-1)|_{x=0}}{4-0}\\\\ &=&\dfrac{(4(4)^3+(4)^2-1)-(-1)}{4}\\\\ &=&68 \end{array}


ii. msec=(4x3+x2−1)∣x=2−(4x3+x2−1)∣x=04−0=(4(2)3+(2)2−1)−(−1)2=16\begin{array}{rcl} m_{sec}&=&\dfrac{(4x^3+x^2-1)|_{x=2}-(4x^3+x^2-1)|_{x=0}}{4-0}\\\\ &=&\dfrac{(4(2)^3+(2)^2-1)-(-1)}{2}\\\\ &=&16 \end{array}


iii. msec=(4x3+x2−1)∣x=4−(4x3+x2−1)∣x=24−2=(4(4)3+(4)2−1)−((4(2)3+(2)2−1)2=118\begin{array}{rcl} m_{sec}&=&\dfrac{(4x^3+x^2-1)|_{x=4}-(4x^3+x^2-1)|_{x=2}}{4-2}\\\\ &=&\dfrac{(4(4)^3+(4)^2-1)-((4(2)^3+(2)^2-1)}{2}\\\\ &=&118 \end{array}



2.
minst=f(2+h)−f(2)(2+h)−2=(4(2+h)3+(2+h)2−1)−(4(2)3+(2)2−1)h=([4h3+24h2+48h+32]+[h2+4h+4]−1−[35])h=4h3+25h2+52hh=4h2+25h+52=4(0.001)2+25(0.001)h+52=52.025\begin{array}{rcl} m_{inst}&=&\dfrac{f(2+h)-f(2)}{(2+h)-2}\\\\ &=&\dfrac{(4(2+h)^3+(2+h)^2-1)-(4(2)^3+(2)^2-1)}{h}\\\\ &=&\dfrac{([4h^3+24h^2+48h+32]+[h^2+4h+4]-1-[35])}{h}\\\\ &=&\dfrac{4h^3+25h^2+52h}{h}\\\\ &=&4h^2+25h+52\\\\ &=&4(0.001)^2+25(0.001)h+52\\\\ &=&52.025 \end{array}

Practice: Rates of Change of Polynomials

Let f(x)=5x3−6x2+2f(x)=5x^3-6x^2+2 be a continuous polynomial function over the interval (−∞,∞).(-\infin,\infin). Determine the following:

Practice: Rates of Change of Polynomials

Let f(x)=4x2−5x+3f(x)=4x^2-5x+3.



Practice: Rates of Change of Polynomials

Let f(x)=±xf(x)=\pm\sqrt{x}.

Estimate the tangent lines at the point x=1x=1 to the nearest hundredth.
(Use h=0.001h=0.001)

Extra Practice
Average & Instantaneous Rate of Change
The position of an object is given by d(t)=3t3−4t+1d(t)=3t^3-4t+1 where dd is in meters and tt is in minutes.
What is the average rate of change between t=1 and t=2? Round to the nearest whole number, do not include units.
Average & Instantaneous Rate of Change
The position of an object is given by d(t)=3t3−4t+1d(t)=3t^3-4t+1 where dd is in meters and tt is in minutes.
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=x2−4f(x)=x^2-4.
Estimate the tangent lines at the x-intercepts to the nearest whole number. (Use h=0.001h=0.001)
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=±−xf(x)=\pm\sqrt{-x}.
Estimate the tangent lines at the point x=−2x=-2. Round to the nearest hundredth. (Use h=0.001h=0.001)
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=3x2+2x−8f(x)=3x^2+2x-8.
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=x3−x+1f(x)=x^3-x+1.
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=7x5−4x+9f(x)=7x^5-4x+9 be a continuous polynomial function over the interval (−∞,∞).(-\infin,\infin). Determine the following:
Rates of Change of Polynomials
Practice: Rates of Change of Polynomials
Let f(x)=2x2+x−5f(x)=2x^2+x-5 be a continuous polynomial function over the interval (−∞,∞).(-\infin,\infin). Determine the following: