Given the function $f(x)=3(x^2-1)$, find the equation of the tangent line at $x=1$.

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Given the function $f(x)=4x^2-1$, calculate the slope of the tangent line at $x=-3$.

Given the function $f(x)=x^2-10x+2$, calculate the slope of the tangent line at $x=2$.

Given the function $f(x)=x^2+100$, calculate the slope of the tangent line at $x=0$.

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) = −0.02x²+50x+100, p(x)=100−0.1x, a=500

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

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) = 200−0.1x, a=1000

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>