Calculus for scientists 1 teacher notes package

Math 1200
Calculus I

Fall 2022
Instructor: Keira Gunn

Lecture Notes



1

,Table of Contents 2. Rules for Differentiation 15

2.0 Learning Outcomes 16
2.1 Derivatives of Polynomials and Exponentials 17
Notation
1. Limits, Continuity, and Rates of Change 4
Constant Rule
1.0 Learning Outcomes 5 Power Rule
1.1 Limits 6 Sum of Functions
Exponentials
Limits
2.2 Product Rule and Quotient Rule 18
Limit laws Product Rule
1.2 Evaluating Limits 7 Quotient Rule
When 𝑓(𝑎) Exists 2.3 Derivatives of Trigonometric Functions 19
Infinite Limits Trigonometric Limits
Evaluating Limits Algebraically Trigonometric Functions
2.4 Chain Rule 20
Squeeze Theorem
Composite Functions
1.3 Continuity 8 Chain Rule
Continuity 2.5 Implicit Differentiation 21
Piecewise Functions 2.6 Inverse Trigonometric Functions and Logarithms 21
Intermediate Value Theorem Inverse Trigonometric Functions
1.4 Limits at Infinity 9 Logarithms
𝑥 Approaching Infinity Logarithmic Differentiation
Asymptotes 2.7 Related Rates 22
Related Rates
1.5 Rates of Change 11
Strategies for Solving Related Rates Problems
Average Rate of Change
Tangent Lines
Instantaneous Rates of Change
1.6 The Derivative 13
The Limit Definition of Derivative
Differentiability
Position, Velocity, and Acceleration
The Derivative as a Function




2

,3. Applications of Derivatives 24 4. Integration 38

3.0 Learning Outcomes 25 4.0 Learning Outcomes 39
3.1 L’Hopital’s Rule 26 4.1 Antiderivatives 40
Indeterminate forms Indefinite Integral
L’Hopital’s Rule Antiderivatives
Applying L’Hopital’s Rule
Initial Value Problems
3.2 Local Extrema 27
Intervals of Increase and Decrease 4.2 Areas Underneath Curves 41
Critical Numbers The Velocity Problem
Local Extrema The Definite Integral
First Derivative Test for Local Extrema Riemmann Sums
3.3 Concavity 29 Computing Areas
The Second Derivative 4.3 The Fundamental Theorem of Calculus 44
Concavity of a Graph
FTC I
Inflection Points
The Second Derivative Test for Local Extrema FTC II
3.4 Graphing 30 Net Change Theorem
3.5 Absolute Extrema 31 4.4 Substitution 46
Absolute Maximum and Minimum The Indefinite Integral
Closed and Bounded Intervals The Definite Integral
Single Critical Point Theorem 4.5 Areas Between Curves 47
3.6 Optimization 32 4.6 Average Value of a Function 48
Strategies for Optimization Problems
Average Value Formula
3.7 Linear Approximation 33
Differentials Mean Value Formula for Integrals
Linear Approximation
3.8 Newton’s Method 34
Newton’s Method
Where Newton’s Method Fails
3.9 Mean Value Theorem 35
Rolle’s Theorem
Upper Bound for the Number of Zeroes
Mean Value Theorem




3

, Section 1


Limits, Continuity, and Rates
of Change


4

, Learning Outcomes

By the end of this section, you will be able to:

 Determine the limit of a function at 𝑥 = 𝑎 by looking at the graph of the function.
 Find the limit of a function at 𝑥 = 𝑎 by analyzing the values of the function for values near 𝑎.
 Compute the limit of a function at 𝑥 = 𝑎 algebraically using the limit laws.
 Compute a limit using the Squeeze Theorem
 Find the discontinuities of a function from its graph.
 Find the discontinuities of a function from its equation using limits.
 Find the discontinuities of a piecewise function.
 Use the Intermediate Value Theorem to approximate the location of a root of a function.
 Use the Intermediate Value Theorem to find a lower bound on the number of roots of a function.
 Compute the limit of a function to infinity algebraically.
 Determine the equations of any asymptotes of a function.
 Compute the average rate of change of a function over a specified interval.
 Compute the instantaneous rate of change of a function at a given point.
 Determine the equation of a tangent line to a function at a given point.
 Compute the derivative of a function at a given point.
 Interpret the derivative of a function as the rate of change.
 Determine if a function is differentiable at a point given its graph.
 Find velocity given a position function.
 Find acceleration given a velocity function.
 Compute the derivative of a function.
 Determine the location of any horizontal tangents to a function.




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