AB CALCULUS
MIDTERM EXAMINATION (at the moment)
24 Multiple Choice Questions
(12 no calculator / 12 with calculator; 3 pts each = 72 pts)
5 Evaluating limits
1 Continuity of piecewise function
4 Finding derivatives (all rules, all methods)
3 Slope of a function at a point
3 Function analysis
1 Inverse of a function
1 Local linearization
1 Average/instantaneous rate of change
1 Composition of functions
1 Odd/even functions
2 Interpret graphs
1 Zeros of a function
EIGHT Free Response Questions
(4 no calculator / 4 with calculator; may have several parts;
12 pts each = 96 pts)
1. Graphical limits
2. Sketching graphs of f(x), f”(x), f”(x)
3. Application of differentiation – optimization & related rate
4. Equations of tangent/normal lines
5. Implicit differentiation
6. Continuity/discontinuity of functions __ and differentiability
7. Graph of function or derivatives to analyze
8. Function analysis, given definition
What we have covered in Differential Calculus:
Unit 1: Limits and Continuity
Average and instantaneous speed
Definition of a limit
One-sided and two-sided limits (and a look at )
Computing limits as ; numerically, graphically, and algebraically
Value of a limit, indeterminate form, and “the limit does not exist”
Properties of limits as
The Squeeze (Sandwich) Theorem and
Horizontal Asymptotes, vertical asymptotes, and infinite limits
Properties of Limits as
Relative rates of growth and dominant functions (i.e. as )
Definition of continuity
Types of discontinuities (hole, jump, asymptote) - removable
Continuous functions and intervals
Properties of continuous functions and related theorems
Intermediate Value Theorem and it’s consequences
Average rate of change and secant slopes
Slope of a curve at a point and tangent (and normal) lines
Free fall investigation
Unit 2: The Derivative
Definition of the derivative for a function and a point
Applications of the derivative definition and approximations
Differentiability and local linearity – basic tangent line problems
Differentiability and continuity
When a function is not differentiable at a point – corners, cusps, and discontinuities
One-sided derivatives and continuous functions on a defined interval
Interpretations of the derivative and notations, and verbal translations (equation verbal)
The second derivative and basic curve sketching
Curve sketching from graphs, equations, tables, characteristics and justification
Rules for differentiation – constant, power, constant multiple, sum and difference - (polynomials)
Exponential function and its derivative
Product and quotient rules
Chain Rule
Trigonometric Functions and their derivatives
Applications of the chain rule – deriving and more
Implicit differentiation
All the rules together
Unit 3 – Applications of the Derivative (2-3 weeks)
Linear approximation and the derivative – tangent line approximations and error
Local linearity to find limits – L’Hopital’s rule – revisit and other cases of the indeterminate form
L’Hopital’s Rule for indeterminate cases
The Mean Value Theorem and Rolle’s Theorem
Return to curve sketching – using the first and second derivatives to interpret and solve application problems
The Extreme Value Theorem
Particle problems (motion along a line) and other applications
Optimization and modeling
Related Rates