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

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

(y - x)² = 2x + 4, (6, 2)

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

x³/² + 2y³/² = 17, (1, 4)

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.

lim (x → 1) (x⁵⁰ − 1) / (x − 1)

For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.

lim (x → −1) (x²/⁹ − 1) / (x + 1)

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 $x=0$.

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

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