(a) How is the number dened (b) Express as a limit. (c) Why is the natural exponential

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)

(a) How is the number dened? (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. When should you use it? (b) Explain how logarithmic differentiation works. When should you use it?

How do you nd the slope of a tangent line to a parametric curve , ?

Write an expression for the linearization of at .

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.d dx 10x  x10x1

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.d dx ln 10  1 10

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.d dx tan2x  d dx sec2x

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.d dx x2  x  2x  1

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. If , then .

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.An equation of the tangent line to the parabola at is .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

Calculate .

If , nd .

If , nd .

If , nd .

Find if .

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.

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.

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

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

(a) If , nd . (b) Find equations of the tangent lines to the curve at the points and . ; (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 , , nd and . ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and .

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

(a) Graph the function in the viewing rectangle by . (b) On which interval is the average rate of change larger: or ?(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 .

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)

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Find in terms of

Find in terms of and .

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

(a) On what interval is the function increasing? (b) On what interval is concave upward?

Find a parabola that passes through the point and whose tangent lines at and have slopes 6 and , respectively.

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?

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.

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 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 cost, in dollars, of producing units of a certain commodity is Cx  920  2x  0.02x2  0.00007x3 (a) Find the marginal cost function. (b) Find and explain its meaning. (c) Compare with the cost of producing the 101st item. (d) For what value of does have an inection point? What is the signicance of this value of ?

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) 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.

A window has the shape of a square surmounted by a semicircle. 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.

Express the limit lim  l  3 cos   0.5    3 as a derivative and thus evaluate it.

Find if it is known that d dx f2x  x2

Evaluate .

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