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Wize AP Calculus (AB) Textbook > Differentiation 1: Basics & Fundamental Properties

Instantaneous Rate of Change & the Tangent Line

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Instantaneous Rate of Change & the Tangent Line


The instantaneous rate of change for some function f(x)f(x) at x=ax=a is a tangent line at x=ax=a and gives the exact rate of change at the value x=a.x=a. It can be expressed as:

Approximate Value of Instantaneous Rate of Change (IRC)=minst=f(a+h)−f(a)h\text{Approximate Value of Instantaneous Rate of Change (IRC)}=m_{inst}=\dfrac{f(a+h)-f(a)}{h}


A tangent line is a line that passes through exactly one point on a curve.

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Example

The population of a group of rabbits is given by P(t)=10,000(1.025)tP(t)=10,000(1.025)^{t}, where P(t)P(t) is the size of the population at time tt in years after the year 2010. Estimate the instantaneous rate of change in the population of rabbits in the year 2025 using:
  1. Centered intervals.
  2. The difference quotient with h=0.001h=0.001.
  3. Using graphing technology.

1.

14.5≤t≤15.5‾:\underline{14.5\leq{t}\leq{15.5}}:

minst=f(x2)−f(x1)x2−x1=10,000(1.025)15.5−10,000(1.025)14.515.5−14.5≈358 rabbits/year\begin{array}{rcl} m_{inst}&=&\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\ &=&\dfrac{10,000(1.025)^{15.5}-10,000(1.025)^{14.5}}{15.5-14.5}\\\\ &\approx&358~\text{rabbits}/\text{year} \end{array}


14.75≤t≤15.25‾:\underline{14.75\leq{t}\leq{15.25}}:

minst=f(x2)−f(x1)x2−x1=10,000(1.025)15.25−10,000(1.025)14.7515.25−14.75≈357 rabbits/year\begin{array}{rcl} m_{inst}&=&\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\ &=&\dfrac{10,000(1.025)^{15.25}-10,000(1.025)^{14.75}}{15.25-14.75}\\\\ &\approx&357~\text{rabbits}/\text{year} \end{array}


2.

minst=f(a+h)−f(a)(a+h)−a=10000(1.025)a+h−10000(1.025)ah=10000(1.02515+0.001−1.02515)0.001≈357 rabbits/year\begin{array}{rcl} m_{inst}&=&\dfrac{f(a+h)-f(a)}{(a+h)-a}\\\\ &=&\dfrac{10000(1.025)^{a+h}-10000(1.025)^a}{h}\\\\ &=&\dfrac{10000\Big(1.025^{15+0.001}-1.025^{15}\Big)}{0.001}\\\\ &\approx&357~\text{rabbits}/\text{year} \end{array}



3. In a graphing calculator:

Y=10000(1.025)x + GRAPH + 2nd + PGRM + 5:Tangent + 15 + ENTER\boxed{Y=10000(1.025)^x}~+~\boxed{\text{GRAPH}}~+~\boxed{2^{\text{nd}}}~+~\boxed{\text{PGRM}}~+~\boxed{5:\text{Tangent}}~+~\boxed{15}~+~\boxed{\text{ENTER}}

The tangent line is approximately y=357x+9118y=357x+9118.

Therefore, the IRC of the population of rabbits is 357 rabbits/ year357~\text{rabbits}/~\text{year}.

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Example: Instantaneous Rate of Change & the Tangent Line

The surface area of a cube can be expressed as SA=6x2SA=6x^2, where xx is the side length measured in centimeters. Estimate the instantaneous rate of change of the surface area of the cube with respect to the side length using the difference quotient when x=4x=4.

ΔVΔx=6(4+h)2−6(4)2(4+h)−4=6(16+8h+h2)−6(16)h=6(16)+48h+6h2−6(16)h=48+6hLet h=0.001=48+6(0.001)=48.006 cm3/cm\begin{array}{rcl} \dfrac{\Delta{}V}{\Delta{}x}&=&\dfrac{6(4+h)^2-6(4)^2}{(4+h)-4}\\\\ &=&\dfrac{6(16+8h+h^2)-6(16)}{h}\\\\ &=&\dfrac{6(16)+48h+6h^2-6(16)}{h}\\\\ &=&48+6h&&\colorTwo{\footnotesize{\text{Let}~h=0.001}}\\\\ &=&48+6(0.001)\\\\ &=&48.006~\text{cm}^3/\text{cm} \end{array}

Practice: Instantaneous Rate of Change & the Tangent Line

Let f(x)=9x3+4f(x)=9x^3+4. Determine the instantaneous rate of change for each interval to the nearest hundredth.

Practice: Instantaneous Rate of Change & the Tangent Line

A rock is shot into the air and the projectile is modelled by the function d(t)=−9.8t2+10t+150d(t)=-9.8t^2+10t+150. Estimate the instantaneous rate of change using the difference quotient at t=3t=3 for the following value of hh. Round answers to the nearest hundredth when required.



Practice: Instantaneous Rate of Change & the Tangent Line

An jug of oil spilled over, leaving a pool of oil on the cement. Estimate how fast the area of the pool of oil is changing using the difference quotient when the radius is 1010cm and allowing h=0.01h=0.01.