Solution: Cobalt-60 has a half-life of 5.24 years. (a) Find

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 . (d) Sketch the graph of

Suppose f is one-to-one, , and . Find (a) and (b) .

Find the inverse function of

Sketch a rough graph of the function without using a calculator. y  5x  1

Sketch a rough graph of the function without using a calculator. y  ex

Sketch a rough graph of the function without using a calculator. y  ln x

Sketch a rough graph of the function without using a calculator. y  lnx  1

Sketch a rough graph of the function without using a calculator. y  2 arctan x

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

Differentiate. y  lnx ln x

Differentiate. y  emx cos nx

Differentiate. y  y  ln sec

Differentiate. y  ln sec x

Differentiate. y  sarctan x

Differentiate. y  x cos1x

Differentiate. f t  t 2 ln t

Differentiate. tt  et 1  et

Differentiate. y  3x ln x

Differentiate. y  cos xx

Differentiate. y  x sinhx2

Differentiate. xey y  x sinhx2  y sin x

Differentiate. h  etan 2

Differentiate. y  arcsin 2x2

Differentiate. y  ln sin x  y  10tan

Differentiate. y  10tan 

Differentiate.  log51  2x

Differentiate. y  ecos x  cosex 

Differentiate. y  y  sin1ex  sx  1 2  x5 x  3

Differentiate. y  sin1ex 

Differentiate. y  x tan14x

Differentiate. y  x 2  14 2x  133x  15

Differentiate. y  lncosh 3x

Differentiate. y  arctan(arcsin sx )

Differentiate. y  cosh1sinh x

Differentiate. y  x tanh1y  cosh1sinh x sx

Differentiate.  cos(estan 3x )

Show that dx 1 2 tan1x  1 4 ln x  12 x2  1  1 1  x1  x2

Find in terms of f x  e f x  tex  tx

Find in terms of f x  tex 

Find in terms of f x  ln tx

Find in terms of f x  tln x

Find f x  2x

Find f x  ln2x

Use mathematical induction to show that if , then

Find an equation of the tangent to the curve at the 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?

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?

Let be the concentration of a drug in the bloodstream. As the body eliminates the drug, decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus , where is a positive number called the elimination constant of the drug. (a) If is the concentration at time , find the concentration at time . (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug?

A cup of hot chocolate has temperature in a room kept at . After half an hour the hot chocolate cools to . (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to ?

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.

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.

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 ?