MATH1010U: Chapter 1
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FUNCTIONS AND MODELS
NOTE: Today is a VERY fast-paced review of some of the pre-calculus knowledge you should have for this course. To get more practice and make sure you understand the concepts, check out the pre-calc packag
MATH1010: Chapter 2
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LIMITS AND DERIVATIVES
Now that we have an understanding of functions, we will move on to study limits of functions, which is the foundation of our future work with derivatives and integrals.
Tangent/Velocity Problems (Sectio
Calculus I Assignment # 1 Question 1
Step 1. We know volume V of a cube of side length x is V=x3. Step 2. Two applications of Pythagora's Theorem on appropriately chosen triangles will give us d2 = x2 + x2 + x2 . Thus, the length of the diagonal, d,
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
Derivatives of Logarithmic Functions (Section 3.8, pg.244)
Question: How do we differentiate logarithmic functions?
Derivative of General Logarithmic Function: Proof:
d 1 (log a x ) = dx x ln
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
Related Rates (Section 3.10 of Stewart, pg. 256)
Recall: Over the past few weeks, weve spent lots of time exploring rules for differentiation. Having learned these rules, lets now turn our atte
MATH1010: Chapter 4 cont
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APPLICATIONS OF DERIVATIVES cont
Maximum and Minimum Values (Section 4.1, pg. 279) cont
Recall: Last day, we looked at max and min values, but we explored this concept graphically.
Example: Sketch a graph of a continuou
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
Related Rates (Section 3.10 of Stewart, pg. 256) cont
Recall: Last day, we looked at several related rates applications.
Example: The combined electrical resistance R of R1 and R2, connected in
MATH1010: Chapter 3 cont and Chapter 4
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DIFFERENTIATION RULES cont
Linear Approximations and Differentials (3.11, pg. 262) cont
Recall: Last day, we introduced the equation for finding a linear approximation.
Example: Find a linearization of f (
MATH1010: Chapter 2 cont.
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LIMITS AND DERIVATIVES cont.
Continuity (Section 2.5 of Stewart, pg. 124)
x + 1 x 3 Question: How does f ( x) = x=3 2 compare with g ( x) = x + 1 ?
Definition: A function f is continuous at a number a if
So, what act
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
The Chain Rule (Section 3.5 of Stewart, pg. 217)
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
MATH1010: Chapter 2 cont
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LIMITS AND DERIVATIVES cont
Tangents, Velocities, and Other Rates of Change (2.7, pg. 149)
Recall: We introduced the limit concept to find slopes of tangent lines. Now that we have techniques for computing limits, lets re
MATH1010: Chapter 2 cont
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LIMITS AND DERIVATIVES cont
Limits at Infinity; Horizontal Asymptotes (Section 2.6, pg. 135)
Recall: Previously, we talked about infinite limits and vertical asymptotes.
Horizontal asymptotes, on the contrary, are based
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Assign #6, Question 1 (Maple version) > f:=x->K*(1+c^2*x^3)/(1+x)^3; f := x K ( 1 + c2 x3 ) ( 1 + x )3
Part a) Find the critical point by setting the derivative to 0. > solve(diff(f(x),x)=0,x); 11 , cc Since x is a ratio of radii, only
MATH1010U: Chapter 3
1
DIFFERENTIATION RULES
Derivatives of Polynomials and Exponential Functions(3.1,p183)
Recall: Last day, we learned how to find the derivative from first principles.
Now, lets consider some useful rules to help us do different
Assignment 3 Solution 1.
a + 0.12 x T ( x) = b + 0.16( x 20000)
lim T ( x) = 0
x 0 +
x 20,000 x > 20,000
i) We want
a +0.12(0) = 0 a=0
lim a +0.12 x = 0 x 0
ii)
0.12 x T ( x) = b + 0.16( x 20000)
x 20,000 x > 20,000
For T(x) to be con
MATH1010U: Chapter 1 cont
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FUNCTIONS AND MODELS cont
Inverse Functions and Logarithms (Section 1.6, pg. 63) cont
Recall: Last class, we reviewed the concept of inverse functions.
Lets now apply this knowledge to find the inverse trigonometric fun
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
Derivatives of Trigonometric Functions (Section 3.4, pg. 211)
In the past few lectures, weve introduced differentiation, and rules to help us take derivatives quickly, but what about trig funct
MATH1010: Chapter 2 cont
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LIMITS AND DERIVATIVES cont
Calculating Limits Using the Limit Laws (2.3, pg. 104)
Recall: Last day, we talked about how to evaluate limits.
Example:
5 lim x 2 cos x 0 x
The Squeeze Theorem: If f ( x) g ( x) h( x)
MATH1010: Chapter 3 cont
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DIFFERENTIATION RULES cont
Implicit Differentiation (Section 3.6 of Stewart, pg.227)
So far, weve always been given y = f (x) explicitly, but what if y is defined implicitly as a function of x? Example: x 2 + y 2 = 1 . F
1. (4 marks) A light is on top of a 15 ft vertical pole. A 5 ft woman walks away from the pole base at a rate of 3 ft/s. How fast is the angle made by the womans head, the light, and the pole base changing 4 seconds after she starts walking? [This qu
MATH1010: Chapter 2 cont
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LIMITS AND DERIVATIVES cont
Precise Definition of a Limit (Section 2.4, pg. 114) cont
2 x + 7, Recall: Last day, we considered the function f ( x) = 5, what it really meant to say that x 1 x =1 and wondered
lim f ( x )