The tangent line approximation

A potato is placed in an oven, and the potato's temperature $F$ (in degrees Fahrenheit) at various points in time is taken and recorded in the following table. Time $t$ is measured in minutes.

1. Use a central difference to estimate $F'(30)$. Use this estimate as needed in subsequent questions.

2. Find the local linearization $y = L(t)$ to the function $y = F(t)$ at the point where $a = 30$.

3. Determine an estimate for $F(33)$ by employing the local…

An object moving along a straight line path has a differentiable position function $y = s(t)$; $s(t)$ measures the object's position relative to the origin at time $t$. It is known that at time $t = 9$ seconds, the object's position is $s(9) = 4$ feet (i.e., 4 feet to the right of the origin). Furthermore, the object's instantaneous velocity at $t = 9$ is $-1.2$ feet per second, and its acceleration at the same instant is $0.08$ feet per secon…

For a certain function $f$, its derivative is known to be $f'(x) = (x-1)e^{-x^2}$. Note that you do not know a formula for $y = f(x)$.

1. At what $x$-value(s) is $f'(x) = 0$? Justify your answer algebraically, but include a graph of $f'$ to support your conclusion.

2. Reasoning graphically, for what intervals of $x$-values is $f''(x) \gt 0$? What does this tell you about the behavior of the original function $f$? Explain.

3. Assuming that $f(2) = -3$, estimate the value of $f(1.88)$ by find…