MATH1020U: Chapter 14 cont
1
PARTIAL DERIVATIVES cont
Directional Derivatives and the Gradient Vector Contd (14.6,
pg.910)
Recall: Last day, we talked about finding partial derivatives in the x and y direction. But
what about a function of more than 2 var
MATH1020U: Chapter 9 cont
DIFFERENTIAL EQUATIONS cont
Recall: Last day, we learned how to solve separable differential equations and set up
simple application problems.
Exponential Growth and Decay (9.4 of Stewart, pg. 591)
NOTE: Some of this material is
MATH1020U: Chapter 8 cont and 9
1
APPLICATIONS OF INTEGRATION cont
Probability (Section 8.5 of Stewart, pg. 555)
Example: Suppose we wanted to know what fraction of students finish their Calc
midterm in less than 50 minhow do we determine this?
Question:
MATH1020U: Chapter 8 cont
1
APPLICATIONS OF INTEGRATION cont
Area of a Surface of Revolution (Section 8.2 of Stewart, pg. 532)
Recall: We just talked about how to find the volume of a solid of revolution. What
about finding its surface area? Well, you can
MATH1020U: Chapter 7 cont and 8
1
TECHNIQUES OF INTEGRATION cont
Reminder: First Midterm Test
When: February 9 and 10, in your lecture section!
Topics: 7.17.8 (excluding 7.6), 8.18.4
[Also, 6.2 and 6.4 will appear in the form of fillintheblank/truef
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Approximate Integration (Section 7.7 of Stewart, pg. 495)
We have now learned several techniques of integration, but do they always work?
Recall: Back in Calculus I, we studied several numerical a
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Strategy for Integration (Section 7.5 of Stewart, pg. 483)
Recall: There are many functions that we already know how to integrate, and sometimes
integrals can be simplified to look like one of the
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Integration of Rational Functions by Partial Fractions (7.4)
3
Recall: We know how to integrate, for example,
But what about
4x + 5
x 2 + x 2 dx
1
x 1 + x + 2 dx
?
Question: Is there any simil
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Substitution (Section 7.3 of Stewart, pg. 489)
Recall: We know how to integrate, for example,
Question: Now what about something such as, e.g.
x
4 x
2
dx
1
4 x
2
dx ?
MATH1020U: Chap
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Integrals (Section 7.2 of Stewart, pg. 460)
Recall: Weve dealt with integrating trigonometric functions before.
Question: Now what about, for example,
Example:
sin 5 x cos x dx
cos
MATH1020U: Chapter 9
1
DIFFERENTIAL EQUATIONS
Modelling with Differential Equations (9.1, pg. 567) cont
Recall: Last day, we introduced the idea of a differential equation.
Recall some more: We had finished off last day talking about the idea of solving a
MATH1020U: Chapter 9 cont
1
DIFFERENTIAL EQUATIONS cont
Separable Equations (Section 9.3 of Stewart, pg. 580)
Recall: Weve just spent time solving differential equations graphically and numerically,
but is it possible to get an exact solution?
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Tutorial 6
The Definite Integral
M129 Applied Calculus
Chapter Outline
Antidifferentiation
Areas
Definite Integrals and the Fundamental Theorem
Areas in the xyPlane
Applications of the Definite Integral
M129 Applied Calculus
Antidifferentiation
Definit
MATH1020U: Chapter 14
1
PARTIAL DERIVATIVES
Functions of Several Variables (Section 14.1 of Stewart, pg.855)
Recall: So far, we have only worked with functions of a single variable. Often,
however, quantities depend on more than one variable.
As in Calcul
MATH1020U: Chapter 15 Contd
1
MULTIPLE INTEGRALS
Double Integrals over General Regions (Section 15.3 of Stewart,
pg.965)
Recall: Last lecture, we looked at double integrals over rectangular regions. The
problem with this is that most of the regions are no
MATH1020U: Chapter 15
1
MULTIPLE INTEGRALS
Double Integrals over Rectangles (Section 15.1 of Stewart,
pg.950)
Recall: In Calculus I, we were interested in finding the area under a curve, and we
approximated this with a sum of rectangles.
Question: How can
MATH1020U: Chapter 10 cont
1
PARAMETRIC EQUATIONS AND POLAR
COORDINATES cont
Polar Coordinates (Section 10.3 of Stewart, pg.639)
Recall: In Calculus I, we studied the unit circle.
y = r sin( )
x = r cos( )
tan( ) =
r2 = x2 + y2
y
x
Example: The Cartesian
MATH1020U: Chapter 10
1
PARAMETRIC EQUATIONS AND POLAR
COORDINATES
Curves Defined by Parametric Equations (Section 10.1, pg. 621)
Recall: A function is a relationship that passes the vertical line test.
Suppose that x and y are both given as functions of
MATH1020U: Calc I Integration Review
1
INTEGRATION REVIEW (from Calculus I)
NOTE: This worksheet works through some basic and more advanced examples
involving usubstitution. We HIGHLY recommend that you take some time to work
through it (and even try the
MATH1010: Chapter 4 cont
1
APPLICATIONS OF DERIVATIVES cont
The Mean Value Theorem (Section 4.2 of Stewart, pg. 280) cont.
Recall: Last class we discussed the Mean Value Theorem and Rolles Theorem.
The Mean Value Theorem: Let f be a function which has th
MATH1010: Chapter 3 cont
1
DIFFERENTIATION RULES cont
Rates of Change in the Natural and Social Sciences (3.7, pg 221)
y f ( x2 ) f ( x1 )
=
is the average rate of change of y with respect to x.
x
x2 x1
dy
y
Also,
is the instantaneous rate of change of
MATH1010: Chapter 3 cont
1
DIFFERENTIATION RULES cont
Implicit Differentiation (Section 3.5 of Stewart, pg.207) cont
Derivatives of Inverse Trigonometric Functions
Question: Earlier in the course, we studied inverse trigonometric functions. How in the wo
MATH1010: Chapter 3 cont
1
DIFFERENTIATION RULES cont
The Chain Rule (Section 3.4 of Stewart, pg. 197)
By now, we know how to easily differentiate a function such as, say, p ( x) = x 7 .
But how do we differentiate something such as h( x) = (5 x 2 + 2) 7
MATH1020U: Chapter 11 cont
1
SEQUENCES AND SERIES cont
Absolute Convergence and the Ratio and Root Tests
(Section 11.6 of Stewart, pg. 714)
Recall: Weve considered convergence and divergence of alternating sequences, but not
yet of series.
a
Definition: