Answer: An equation of motion of the form represents

The graph of is shown. Is one-to-one? Explain.

The graph of is given. (a) Why is one-to-one? (b) Estimate the value of . (c) Estimate the domain of .

Suppose f is one-to-one, , and . Find (a) and (b) . g y 0 1 x 1 t1 f 7

Find the inverse function of

Sketch a rough graph of the function without using a calculator

Sketch a rough graph of the function without using a calculator

Sketch a rough graph of the function without using a calculator

Sketch a rough graph of the function without using a calculator

Sketch a rough graph of the function without using a calculator

Let . For large values of , which of the functions , , and has the largest values and which has the smallest values?

Find the exact value of each expression.

Find the exact value of each expression.

Solve the equation for .

Solve the equation for .

Solve the equation for .

Solve the equation for .

Solve the equation for .

Solve the equation for .

Solve the equation for .

Solve the equation for .

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Differentiate.

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Use mathematical induction to show that if , then

Find an equation of the tangent to the curve at the given point.

Find an equation of the tangent to the curve at the given point.

At what point on the curve is the tangent horizontal?

If , find . Graph and on the same screen and comment.

(a) Find an equation of the tangent to the curve that is parallel to the line . (b) Find an equation of the tangent to the curve that passes through the origin.

The function , where a, b, and K are positive constants and , is used to model the concentration at time t of a drug injected into the bloodstream. (a) Show that . (b) Find , the rate at which the drug is cleared from circulation. (c) When is this rate equal to 0?

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Evaluate the limit.

Sketch the curve using the guidelines of Section 3.5.

Sketch the curve using the guidelines of Section 3.5.

Sketch the curve using the guidelines of Section 3.5.

Sketch the curve using the guidelines of Section 3.5.

Sketch the curve using the guidelines of Section 3.5.

Sketch the curve using the guidelines of Section 3.5.

Investigate the family of curves given by , where is a real number. Start by computing the limits as . Identify any transitional values of where the basic shape changes. What happens to the maximum or minimum points and inflection points as changes? Illustrate by graphing several members of the family

Investigate the family of functions . What happens to the maximum and minimum points and the inflection points as changes? Illustrate your conclusions by graphing several members of the family

An equation of motion of the form represents damped oscillation of an object. Find the velocity and acceleration of the object.

a) Show that there is exactly one root of the equation and that it lies between 2 and . (b) Find the root of the equation in part (a) correct to four decimal places.

A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000?

Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?

The biologist G. F. Gause conducted an experiment in the 1930s with the protozoan Paramecium and used the population function to model his data, where was measured in days. Use this model to determine when the population was increasing most rapidly.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Use properties of integrals to prove the inequality.

Use properties of integrals to prove the inequality.

Use properties of integrals to prove the inequality.

Find the average value of the function on the interval

Find the area of the region bounded by the curves , and

Find the volume of the solid obtained by rotating about the -axis the region under the curve from to

What is the area of the largest rectangle in the first quadrant with two sides on the axes and one vertex on the curve ?

What is the area of the largest triangle in the first quadrant with two sides on the axes and the third side tangent to the curve ?

Evaluate without using the Fundamental Theorem of Calculus. [Hint: Use the definition of a definite integral with right endpoints, sum a geometric series, and then use lHospitals Rule.]

If , where , then, by the Fundamental Theorem,

Show that cos arctansinarccot x x 2 1 x 2 2 F1

If is a continuous function such that for all , find an explicit formula for

The figure shows two regions in the first quadrant: is the area under the curve from to , and is the area of the triangle with vertices , , and . Find . At y sinx 0 t Bt 2 PO t, 0 lim tl 0 AtBt O y xt y=