2. Intro to Derivatives

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.

d. During what year did home prices drop most rapidly and what is an estimate of this rate?

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km / q) + cm + (hq / 2),

where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

Find dA/dq and d²A/dq².

Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

<IMAGE>

d. $p^{\prime }\left(2\right)$

Derivative of y = |x| Graph the derivative of f(x) = |x|. Then graph y = (|x| − 0)/(x − 0) = |x|/x. What can you conclude?

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?

In Exercises 65 and 66, find the derivative using the definition.

ƒ(t) = 1 .

2t + 1

{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by

f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>

Let f (x) = √x.

a. Find the exact value of f' (4).

In Exercises 19–22, find the values of the derivatives.

dr/dθ |θ₌₀ if r = 2/√(4 – θ)

For x < 0, what is f′(x)?

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(t) = 3t⁴; a= -2, 2

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).

" style="" width="350">

" style="" width="200">

Find the derivative of the function $f(x)=4x^2-9x$.

Graph the function $f\left(x\right)=\{\begin{array}{c}x\text{ if }x\le 0\\ x+1\text{ if }x>0\end{array}$.