Linearization (College Board AP® Calculus AB): Exam Questions

Find the linear approximation to $f\left(x\right)=\sqrt[3]{x}$ at $x=\frac{27}{8}$.

Use the linear approximation found in part (a) to approximate the value of $\sqrt[3]{4}$. Find the percentage error of the approximation compared to the accurate value.

A hot beverage is left in a room to cool, and the temperature of the beverage is modeled by:

$T\left(t\right)=85{\left(0.97\right)}^t+75$

Where $T\left(t\right)$ is the temperature of the beverage in degrees Fahrenheit, and $t$ is the time in minutes since the beverage was left to cool.

For $t>30$, the linear approximation $L\left(t\right)$ to $T\left(t\right)$ at $t=30$ is a better model for the temperature of the beverage.

Use $L\left(t\right)$ to predict the time, to the nearest second, at which the temperature of the beverage will reach 90°F. Show the work that leads to your answer.

Consider the differential equation $\frac{dy}{dx}=x^2{y}^2$.

Let $y=f\left(x\right)$ be a particular solution to the differential equation with $f\left(2\right)=3$.

Use the tangent line equation at $x=2$ to approximate $f\left(1.9\right)$.

Solutions to the differential equation in part (a) also satisfy $\frac{d^{2}y}{d{x}^2}=2x{y}^2\left(1+x^{3}y\right)$.

Determine whether the approximation for $f\left(1.9\right)$ from part (a) is an overestimate or an underestimate. Explain your reasoning.

The population of a rare bird species in a wildlife reserve is measured in hundreds of birds and is modeled by a twice-differentiable function $P\left(t\right)$, where $t$ is the number of years since the species was first introduced to the reserve.

The table below gives selected values of the rate of change, $P\text{'}\left(t\right)$, of the population over the time interval $0\le t\le 15$. The population is 40 000 birds when $t=8$.

The graph of $P\left(t\right)$ is known to be concave down for $8\le t\le 10$.

Estimate the population of the bird species at $t=8.5$ using the tangent line approximation at $t=8$. Is your estimate greater than or less than the true value of $P\left(t\right)$? Justify your answer.

Let $f$ be a function that is differentiable for all real numbers. The table below gives the values of $f$ and its derivative $f\text{'}$ for selected values of $x$ in the closed interval $-1.5\le x\le 1.5$. The second derivative of $f$ has the property that $f\text{'}\text{'}\left(x\right)>0$ for $1\le x\le 1.5$.

Write an equation of the line tangent to the graph of $f$ at the point where $x=1$. Use this line to approximate the value of $f\left(1.3\right)$. Is this approximation greater than or less than the actual value of $f\left(1.3\right)$? Give a reason for your answer.

Consider the curve defined by the equation $x^2+3y+2y^2=48$. It can be shown that $\frac{\text{d}y}{\text{d}x}=\frac{-2x}{3+4y}$.

There is a point on the curve near $(2,4)$ with $x$-coordinate 3. Use the line tangent to the curve at $(2,4)$ to approximate the $y$-coordinate of this point.

Consider the differential equation $\frac{dy}{dx}=\frac{1}{3}\cos \left(\frac{\pi }{3}x\right)\sqrt{y+9}$. Let $y=f\left(x\right)$ be the particular solution to the differential equation with the initial condition $f\left(1\right)=7$.

Write an equation for the line tangent to the graph of $y=f\left(x\right)$ at the point (1, 7). Use the equation to approximate $f\left(0.5\right)$.

It is known that $f^{\text{'}\text{'}}\left(x\right)<0$ for $0\le x\le 1$. Is the approximation found in part (a) an overestimate or an underestimate for $f\left(0.5\right)$? Give a reason for your answer.

Use a linear approximation for $f\left(x\right)=\ln x$ at $x=e$ to show that $\frac{a}{e}$, where $a$ is a constant, is an approximation for $\ln 3$.

The function $\theta \left(t\right)$ describes the measured temperature, $\theta$, of a substance in degrees Celsius ($°\mathrm{C}$), where $t$ is the time in minutes since the substance has been removed from a refrigerator.

The rate of change of $\theta$ with respect to time is given by the differential equation $\frac{d\theta }{dt}=\frac{1}{5}\left(25-\theta \right)$ and it is known that the temperature of the substance at $t=0$ was $5°\mathrm{C}$.

Use the line tangent to the graph of $\theta$ at $t=0$ to approximate $\theta \left(1.5\right)$, the temperature of the substance at time $t=1.5$ minutes.

Write an expression for $\frac{d^2\theta }{d{t}^2}$ in terms of $\theta$. Use $\frac{d^2\theta }{d{t}^2}$ to determine whether the approximation from part (a) is an underestimate or overestimate for the actual value of $\theta \left(1.5\right)$. Give a reason for your answer.