?The function , where a, b, and K are positive constants and , is used to model the

State each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f ) The Quotient Rule (g) The Chain Rule

State the derivative of each function. (a) (b) (c) (d) (e) (f ) (g) (h) (i) ( j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t)

(a) How is the number defined? (b) Express as a limit. (c) Why is the natural exponential function used more often in calculus than the other exponential functions ? (d) Why is the natural logarithmic function used more often in calculus than the other logarithmic functions ?

(a) Explain how implicit differentiation works. (b) Explain how logarithmic differentiation works.

Give several examples of how the derivative can be interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences.

(a) Write a differential equation that expresses the law of natural growth. (b) Under what circumstances is this an appropriate model for population growth? (c) What are the solutions of this equation?

(a) Write an expression for the linearization of at . (b) If , write an expression for the differential . (c) If , draw a picture showing the geometric meanings of and .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If and are differentiable, then

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 2. If and are differentiable, then

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 3. If and are differentiable, then

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 4. If is differentiable, then .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 5. If is differentiable, then .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. If , then .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 7.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 8.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 9.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 10.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 11. The derivative of a polynomial is a polynomial.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 12. If , then .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 13. The derivative of a rational function is a rational function.

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 14. An equation of the tangent line to the parabola at is .

1-15 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 15. If , then

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If , find .

If , find .

Find if +.

Find if .

Use mathematical induction (page 76) to show that if , then .

Evaluate .

57–59 Find an equation of the tangent to the curve at the given point. 57.

57–59 Find an equation of the tangent to the curve at the given point. 58.

57–59 Find an equation of the tangent to the curve at the given point. 59.

60–61 Find equations of the tangent line and normal line to the curve at the given point. 60.

60–61 Find equations of the tangent line and normal line to the curve at the given point. 61.

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

(a)If , find . (b) Find equations of the tangent lines to the curve at the points (1, 2) and (4, 4). (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .

(a) If , , find and . (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and .

At what points on the curve , , is the tangent line horizontal?

Find the points on the ellipse where the tangent line has slope 1.

If , show that

(a) By differentiating the double-angle formula obtain the double-angle formula for the sine function. (b) By differentiating the addition formula obtain the addition formula for the cosine function.

Suppose that and , where , , , , and . Find (a) and (b) .

If and are the functions whose graphs are shown, let , , and . Find (a) , (b) , and (c) .

71–78 Find in terms of . 71.

71–78 Find in terms of . 72.

71–78 Find in terms of . 73.

71–78 Find in terms of . 74.

71–78 Find in terms of . 75.

71–78 Find in terms of . 76.

71–78 Find in terms of . 77.

71–78 Find in terms of . 78.

79–81 Find in terms of and . 79.

79–81 Find in terms of and . 80.

79–81 Find in terms of and . 81.

(a) Graph the function in the viewing rectangle by . (b) On which interval is the average rate of change larger: [1, 2] or [2, 3]? (c) At which value of is the instantaneous rate of change larger: or ? (d) Check your visual estimates in part (c) by computing and comparing the numerical values of and .

At what point on the curve is the tangent horizontal?

(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.

Find a parabola that passes through the point (1, 4) and whose tangent lines at and have slopes 6 and -2, respectively.

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?

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

A particle moves along a horizontal line so that its coordinate at time is , , where and are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction.

A particle moves on a vertical line so that its coordinate at time is , . (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval . (d) Graph the position, velocity, and acceleration functions for . (e) When is the particle speeding up? When is it slowing down?

The volume of a right circular cone is , where is the radius of the base and is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant.

The mass of part of a wire is kilograms, where is measured in meters from one end of the wire. Find the linear density of the wire when m.

The cost, in dollars, of producing units of a certain commodity os (a) Find the marginal cost function. (b) Find and explain its meaning. (c) Compare with the cost of producing the 101st item.

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 t 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 80°C in a room kept at 20°C. After half an hour the hot chocolate cools to 60°C. (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to 40°C?

The volume of a cube is increasing at a rate of 10 . How fast is the surface area increasing when the length of an edge is 30 cm?

A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of , how fast is the water level rising when the water is 5 cm deep?

A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?

A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp?

The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is ?

(a) Find the linear approximation to near 3. (b) Illustrate part (a) by graphing and the linear approximation. (c) For what values of is the linear approximation accurate to within 0.1?

(a) Find the linearization of at . State the corresponding linear approximation and use it to give an approximate value for . (b) Determine the values of for which the linear approximation given in part (a) is accurate to within 0.1.

Evaluate if , , and .

A window has the shape of a square surmounted by a semi-circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.

106–108 Express the limit as a derivative and evaluate. 106.

106–108 Express the limit as a derivative and evaluate. 107.

106–108 Express the limit as a derivative and evaluate. 108.

Evaluate .

Suppose is a differentiable function such that and . Show that .

Find if it is known that

Show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant.