2. Intro to Derivatives

Derivatives and tangent lines

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

f(x) = 8x; a = −3

Find an equation of the line tangent to the following curves at the given value of x.

y = 4 sin x cos x; x = π/3

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

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

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 $f(x)=3(x^2-1)$, 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

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>

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.

c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b). 

{Use of Tech} Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W=1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh=3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t²− t³/9 kWh where t=0 corresponds to midnight.

a. Graph the energy function. 

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

b. How long does it take for the tank to empty? 

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

a. Graph P using a graphing utility. Experiment with different windows until you produce an S-shaped curve characteristic of the logistic model. What window works well for this function?