Exam 2: Derivatives

Find the differential of the function at the indicated number.

Find the derivative of the function. g(t) = tan(cos 2t)

Find equations of the tangent lines to the curve that are parallel to the line .

The sides of a square baseball diamond are 90 ft long. When a player who is between the second and third base is 30 ft from second base and heading toward third base at a speed of 24 ft/sec, how fast is the distance between the player and home plate changing? Round to two decimal places.

Find the derivative of the function.

The altitude of a triangle is increasing at a rate of while the area of the triangle is increasing at a rate of . At what rate is the base of the triangle changing when the altitude is 10 cm and the area is .

Find by implicit differentiation.

In this exercise we estimate the rate at which the total personal income is rising in the Richmond- Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,600, and the population was increasing at roughly 9,400 people per year. The average annual income was $30,591 per capita, and this average was increasing at about $1,300 per year (a little above the national average of about $1,225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999.

Find the derivative of the function and evaluate at the given value of x.

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁷;

Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\pi$ /3 .

Find the derivative of the function.

Identify the "inside function" u = f (x) and the "outside function" y = g (u). Then find dy/dx using the Chain Rule.

Suppose that Find

Find the derivative of the function.

Find the limit.

The position function of a body moving along a coordinate line is s(t) = 3 sin t + 4 cos t where t is measured in seconds and s(t) in feet. Find the position, velocity, speed, and acceleration of the body when t = .

Find the derivative of the function.

Use the table to estimate the value of , where and . 10 10.1 10.2 10.3 10.4 10.5 10.6 4.5 5.6 4.3 2.5 9.9 7.8 6.5 5.9 4.7 4.2 5.4 6.3

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. Show that the curves of the given equations are orthogonal. y - x = x = cos y