2. Intro to Derivatives

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

f(x) = -7x; P(-1,7)

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

f(x) = 1/x; P (1,1)

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x-2)^1/3

Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y = 2/x; P(1, 2)

Derivatives and tangent lines

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

f(x) = √3x; a= 12

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y= √x; P(4, 2)

In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.

73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2

{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. 

In Exercises 19–22, find the slope of the curve at the point indicated.

y = (x − 1) / (x + 1), x = 0

Interpreting Derivative Values

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.

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a. Explain what is meant by the derivative P'(5). What are its units?

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)=x²+1; Q(3, 6)

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? 

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

f(x)= 1/(2x + 1); P (0,1)

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

f(x) = 2/√x; P(4,1)