Original stupidly long list of Module Level Objectives for MTH 201
Module 1: How do we measure velocity? (1.1, 1.2)
- Compute the average velocity of a function on an interval using either of the average velocity formulas.
- Explain the differences between average velocity and instantaneous velocity.
- Find the instantanous velocity of a moving object through a sequence of average velocities.
- Explain the notation used for limits.
- Find the limit of a function as the input approaches a point, using tables and graphs.
- Find the limit of a function as the input approaches a point, using algebraic simplification.
- Find the instantanous velocity of a moving object by setting up and computing a limit.
Module 2: What is a derivative? (1.3, 1.4)
- State the formal definition of the derivative of a function on an interval.
- Interpret the meaning of a derivative in terms of instantaneous velocity and the slopes of tangent and secant lines.
- Compute the derivative of a function at a point by setting up and evaluating the limit in its definition.
- State the formal definition of the derivative of a function.
- Given the graph of a function, make a reasonable sketch of the graph of its derivative.
- Compute the formula for the derivative of a function by setting up and evaluating the limit in its definition.
Module 3: What information does the derivative give us? (1.5, 1.6)
- If a function is given with units for the input and output, state the units of the derivative.
- Estimate the value of a derivative at a point using forward, backward, and central difference calculations.
- Interpret the meaning of the derivative of a function that models a real-life situation, in terms instantaneous rates of change.
- Determine whether a function is increasing, decreasing, or constant on an interval by examining the sign of its derivative. (Conversely, state the sign of a function's derivative using information about whether the function is increasing, decreasing, or constant.)
- State the definition of the second derivative of a function.
- Explain the meaning of the terms concave up and concave down and explain the connection between concavity of a function and its first and second derivatives.
- Determine whether a function is concave up or concave down on an interval by examining its graph.
- Given the graph of a function, make a reasonable sketch of the graph of its second derivative.
- If a function is given with units for the input and output, state the units of the second derivative.
Module 4: Does every function have a derivative? (1.7, 1.8) -- 9/21-9/17
- Compute left- and right-hand limits of a function at a point.
- Describe the conditions under which the limit of a function at a point exists; and state the conditions under which a limit will fail to exist.
- State and explain the conditions under which a function is continuous (or fails to be continuous) at a point or on an interval.
- State and explain conditions under which a function is differentiable at a point (or fails to be differentiable at a point).
- Determine whether a function is continuous at a point or differentiable at a point given its graph or a formula.
- Find the local linearization of a function near a given point, and use the local linearization to predict values of the function.
- Use the concavity of a function to determine whether a local linear approximation is an overestimate or underestimate.
Module 5: How do we compute derivatives of basic functions? (2.1, 2.2) -- 9/28-10/1
- Compute (without using limits) the derivatives of constant, power, and exponential functions.
- Compute the derivative of constant multiples and sums of constant, power, and exponential functions.
- Compute the derivative of the sine and cosine functions.
- Use basic derivative rules to solve problems about slopes, velocities, and rates of change involving basic functions.
Module 6: How do we compute derivatives of products and quotients? (2.3, 2.4) -- 10/5-10/8
- State the Product Rule and Quotient Rule.
- Use the Product Rule to differentiate products of basic functions.
- Use the Quotient Rule to differentiate quotients of basic functions.
- Compute the derivatives of the tangent, cotangent, secant, and cosecant functions.
Module 7: How do we compute derivatives of composites? (2.5, 2.6) -- 10/12-10/15
- State the Chain Rule.
- Use the Chain Rule to differentiate a composite of two basic functions.
- Differentiate a function whose derivative involves a mixture of rules (Product, Quotient, Chain, etc.)
- Compute the derivative of the function $y = \ln(x)$.
- Compute the derivatives of the arcsine and arctangent functions.
- Given an invertible function, state a rule for the derivative of its inverse.
Module 8: How do we use derivatives to work with implicit curves and indeterminate limits? (2.7, 2.8) -- 10/19-10/22
- Explain what it means for a function to be an implicit function of a variable.
- Given an expression in which a variable $y$ is implicitly a function of another variable $x$, compute $dy/dx$.
- Find the slope or equation of a tangent line to a curve given implicitly by an expression.
- State whether a given limit is an indeterminate form, and if so, the state the type of indeterminate form.
- State L'Hôpital's Rule.
- Use L'Hôpital's Rule to evaluate limits of an indeterminate form.
- Compute the limit of a function as the input increases or decreases without bound.
Module 9: How can derivatives tell us important information about functions? (3.1, 3.3) -- 10/28-11/3 (begins to wrap around weekends)
Identify the global and local minimum and maximum values of a function, given the function's graph.
State the definition of critical number and identify all the critical numbers of a function given the function's graph.
Find all the critical numbers of a function algebraically using derivatives.
State the First Derivative Test and the Second Derivative Test.
Use the First Derivative Test to determine whether a critical number of a function is a relative minimum, relative maximum, or neither.
Use the Second Derivative Test to determine whether a critical number of a function is a relative minimum or relative maximum.
Identify the inflection points of a function, given the function's graph.
Use the second derivative to find all inflection points of a function algebraically.
Use the Extreme Value Theorem to find the absolute maximum and minimum values of a continuous function on a closed interval.
Module 10: How can we use derivatives to find the best value of a function? (3.3, 3.4) -- 11/4-11/10
- Set up and solve applied optimization problems.
- Set up and solve related rates problems.
Module 11: How to we tell how far something has traveled if all we know is its speed? (4.1, 4.2) -- 11 11-11/17
- Use area to estimate or compute the distance traveled by an object, given a graph of its velocity.
- Determine whether one function is an antiderivative of another.
- Use an antiderivative to compute the distance traveled by an object, given a formula for its velocity.
- Estimate the area between a curve and the horizontal axis on an interval using a left, right, and middle Riemann sum.
Module 12: What is a definite integral, and how is it related to a derivative? (4.3, 4.4) -- 11/18-11/24 (back to MTWR)
- State the formal definition of the definite integral of a function on an interval and explain the meaning of each part of the notation.
- Find the exact value of a definite integral using known geometric formulas (if possible).
- Use the Properties of the Definite Integral to compute, manipulate, and simplify definite integrals.
- Calculate the average value of a function on an interval.
- State the Fundamental Theorem of Calculus.
- Find the most general antiderivative of a function.
- Use the Fundamental Theorem of Calculus to find the exact value of a definite integral using an antiderivative.
- Find the total amount of change in a function on an interval, using a definite integral.