2. Intro to Derivatives

Find the derivative function f' for the following functions f.

f(x) = 2/3x+1; a= -1

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x + 3); P (1,2)

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 3x² - 4x; P(1, -1)

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

f(x) = √(x + 1), (8, 3)

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/ x²; a= 1

If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.

U.S. population growth The population p(t) (in millions) of the United States t years after the year 1900 is shown in the figure. Approximately when (in what year) was the U.S. population growing most slowly between 1925 and 2020? Estimate the growth rate in that year. <IMAGE>

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x) = 1/x; Q (-2, 4)

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x² - 5x + 1; P(1,-7)

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = x²; a=3

Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = xp(c) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item, given that x items have already been sold. Consider the following cost functions C and price functions p.

a. Find the profit function P.

C(x) = − 0.04x²+100x+800, p(x)=200, a=1000

Find the equation of the tangent line to the curve $y=\sin \left(x\right)+\cos \left(x\right)$ at the point $(0,1)$.

Analyzing Motion Using Graphs

[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:

e. When is it moving fastest (highest speed)? Slowest?

s = 4 - 7t + 6t² - t³, 0 ≤ t ≤ 4