Read the Example of Sec. 7.5, staring at the bottom of p. 328.

Solve, write-up and turn in: Sec. 7.4: 3, 7, 18 where take the formula for the parabola y for granted and solve (a), (b).

Sec. 7.5: 1, 3, 5, 11

Topics: 7.6

Solve, write-up and turn in: Sec. 7.6: 1, 3, 7, 9, 11. Bonus: 12.

Topics: 8.2, start 8.4

Solve, write-up and turn in: 1. We defined ln x for x>0 by ln(x) = the definite integral, from 1 to x, of 1/t dt.

What about the definite integral of 1/t dt on an interval included in the negative x-axis? Show that

the integral, from -1 to x (where x<0), of 1/t dt equals ln|x|+a constant.

2. Using the definition of ln(x) as the definite integral, from 1 to x, of 1/t dt show that ln(1/x)=-ln(x).

Hint: substitute t=s/x (s is the new variable of integration).

3. Furthermore, show that ln(xy)=ln(x)+ln(y).

Hint: substitute t=sy (s is the new variable of integration) and use 2. above.

Topics: 8.2-8.4, review the inverse of a function

For Exam II: you need to know the graph and properties of ln(x) and of exp(x), and to integrate and differentiate these functions.

Solve: Sec. 8.4: 1e-o,

Solve, write-up and turn in: Sec. 8.3: 1, 3 , 11, 15, 19, 24, 25, 27

Sec. 8.4: 1a,b,c,d, 2a,b,c, 3a, 5a, b,k, 17

Solve for review (you do not need to turn them in): Sec. 8.4: 8-12

Review topics for Exam II

Related rates

Integration by substitution (5.3)

Differential equations (5.4)

Area under the graph of a positive function. Riemann sums.

Definition of a Riemann integrable function on [a,b].

Theorem: a continuous function on a closed interval is Riemann integrable.

Simple examples of functions not continuous, yet integrable.

The definite integral of f(x) on [a,b] (the Riemann integral) is the area between the graph of f and the segment [a,b] with sign.

Calculation of areas as limits of Riemann sums.

And the other way around: calculation of limits of sums noting that they are Riemann sums.

The Fundamental Theorem of Calculus.

Properties of definite integrals.

Area between two curves.

Volumes of solids of revolution: disk (or washer) method, cylindrical shells method.

Volumes by integrating the areas of the cross-sections.

Arc length.

Area of a surface of revolution.

e^x and ln: they are inverse to each other, limits for e, their domains, range, derivatives, graphs,

integrating 1/x, and e^x (as such or in combination with substitutions)

Make sure you know how to solve all the assigned homework problems.