Exam 3: Applications of Differentiation

Suppose the line is tangent to the curve when . If Newton's method is used to locate a root of the equation and the initial approximation is , find the second approximation .

For the function f whose graph is given, find the limit.

Sketch the graph of the function using the curve-sketching guidelines.

The graph of the derivative of a continuous function f is shown. On what intervals is f decreasing?

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.

Given . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of F.

The sum of two positive numbers is . What is the smallest possible value of the sum of their squares?

Use Newton's method to approximate the zero of between and using . Continue until two successive approximations differ by less than 0.00001.

Find the critical numbers of .

Given . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

The graph of the first derivative of a function f is shown below. At what values of x does f have a local maximum or minimum?

Find the slant asymptote of the function .

Find the position function of a particle moving along a coordinate line that satisfies the given condition. , s(1) = -1

Sketch the graph of the function using the curve-sketching guidelines.

Let f (x) = 4x³ - 5x² + x - 2 and g (x) = 4x³ a.Plot the graph of f and g using the same viewing window: [-5, 5] ´ [-5, 5]. b.Plot the graph of f and g using the same viewing window: [-50, 50] ´ [-100,000, 100,000] c.Explain why the graphs of f and g that you obtained in part (b) seem coalesce as x increases or decreases without bound.

Sketch the graph of on and find its absolute maximum and absolute minimum values, if any.

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve at some point?

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function.

The function satisfies the hypotheses of the Mean Value Theorem on the interval . Find all values of c that satisfy the conclusion of the theorem.

A company estimates that the marginal cost (in dollars per item) of producing items is . If the cost of producing one item is $560 find the cost of producing 400 items.