What are examples you can give to Calculus I high school students?
Here is a link to show the level linear approximations will be taught.
I've found some applications by a simple google search. Mostly, they deal with physics, and it seems that the idea is to make a function easier to deal with since the linear approximation is so close to the actual value.
I have a thermal expansion problem I used last year: $\frac{dP}{dT} = kP$ where $P = 15$cm when $T = 25℃$, and they needed to find the change in length of the cable when $T = 28℃$.
Also, $k = 1.7 × 10^{−5}℃^{−1}$.
Mostly, I am asking this question to see if some of you have better examples and/or explanations. I can keep searching for more examples, but I am not very familiar with physics and know that I probably won't pick the best example to illustrate how linear approximation can be useful.
Best Answer
What are some applications of linear approximation in the real world?
I present below a compilation of possibilities. You can find more applications and more details in the mentioned books (as well as in similar books).
Solution: Taking $f(x)=x^4$ and $a=2$ in the formula $$L(x)=f(a)+f'(a)(x-a),$$ we obtain $$L(x)=32x-48.$$ Therefore, $$(1.999)^4=f(1.999)\approx L(1.999)=32(1.999)-48=15.968.$$
Solution: Taking $f(x)=6x^2$ and $a=30$ in the previous formula, we obtain $$L(x)=f(30)+f'(30)(x-30)$$ Therefore, $$\begin{aligned} \text{Maximum possible error}&=f(30\pm 0.1)-f(30)\\ &\approx L(30\pm 0.1)-f(30)\\ &=f'(30)(\pm 0.1)\\ &=\pm 36\;\text{cm}^2 \end{aligned}$$
Solution: Taking $f(x)=\tfrac{2}{3}\pi x^3 $ and $a=25$ in the previous formula, we obtain $$L(x)=f(50)+f'(25)(x-25)$$ Therefore, $$\begin{aligned} \text{Amount of paint}&=f(25.0005)-f(25)\\ &\approx L(25.0005)-f(25)\\ &=f'(25)(0.0005)\\ &=0.625\pi\\ &\approx 1.96\;\text{m}^3 \end{aligned}$$
Applications to Physics
Ohm’s Law (Larson's book). A current of $I$ amperes passes through a resistor of $R$ ohms. Ohm’s Law states that the voltage $E$ applied to the resistor is $E=IR$. If the voltage is constant, show that the magnitude of the relative error in $R$ caused by a change in $I$ is equal in magnitude to the relative error in $I$.
Projectile Motion (Larson's book). The range of a projectile is $R=\frac{v_0^2}{32}\sin (2\theta)$ where $v_0$ is the initial velocity in feet per second and $\theta$ is the angle of elevation. If $v_0=2200$ feet per second and $\theta$ is changed from $10^\circ$ to $11^\circ$ , use linear approximation to approximate the change in the range.
Period of a pendulum (Anton's book). The time required for one complete oscillation of a pendulum is called its period. If $L$ is the length of the pendulum and the oscillation is small, then the period is given by $P =2\pi\sqrt{\frac{L}{g}}$, where $g$ is the constant acceleration due to gravity. Show that the percentage error in $P$ is approximately half the percentage error in $L$.
Richter scale (Anton's book). The magnitude $R$ of an earthquake on the Richter scale is related to the amplitude $A$ of the shock wave by the equation $R=\log_{10}(\frac{A}{A_0})$, where $A_0$ is a small positive constant. (Originally, $A_0$ was the smallest possible amplitude that could be detected.) Show that for a small change in $A$ the change in $R$ can be approximated by multiplying the relative change $\Delta A/A$ in amplitude by $0.4343$.
The small angle approximation (Anton's book).