MATH 181 Calculus I

Course Learning Objectives by Chapter and Section (Exercises)

UNIT I

2.1 The Idea of Limits

a)                  Calculate average velocity from a given distance function. (1,3,7,24)

b)                  Estimate instantaneous velocity numerically. (9,15,24)

2.2 Definitions of Limits

a)                  Understand limit definitions. (1,3,5)

b)                  Find limits from a graph. (7,17,19,25)

c)                  Estimate limits from a table. (13,21)

d)                  Analyze limits numerically. (37,43)

2.3 Techniques for Computing Limits

a)                  Apply limit laws. (20,55)

b)                  Evaluate limits analytically. (24,27,39,44,45)

c)                  Evaluate one-sided limits. (31,33)

d)                  Use squeeze theorem to find limits. (51)

2.4 Infinite Limits

a)                  Find infinite limits numerically or graphically. (10,15)

b)                  Evaluate infinite limits analytically. (17,20,30)

c)                  Find vertical asymptotes. (23)

2.5 Limits at Infinity

a)                  Understand end behavior and horizontal asymptotes. (3,5)

b)                  Evaluate limits at infinity. (12, 17)

c)                  Find horizontal and vertical asymptotes. (24,45)

d)                  Determine end behavior and sketch graphs. (31)

e)                  Use limits to find steady states in applications. (57)

2.6 Continuity

a)                  Understand continuity. (2,5,73)

b)                  Find points of discontinuity or intervals of continuity. (11,15,17,22,27,33,35)

c)                  Evaluate limits using principles of continuity. (45,46)

d)                  Use the Intermediate Value Theorem. (51,80)

2.7 Precise Definitions of Limits

a)                  Determine delta from the graph of a function, for given epsilon. (9,10,12)

UNIT II

3.1 Introducing the Derivative

a)                  Understand derivatives graphically. (39,41,43,53,56)

b)                  Understand differentiability and relate it to continuity. (9,46)

c)                  Evaluate derivatives and work with equations of tangent lines. (11,15,22,50)

3.2 Rules of Differentiation

a)                  Find derivatives of constant multiples of powers and sums of functions. (7,10,13,23)

b)                  Simplify products and quotients and take their derivatives. (26,31)

c)                  Use derivatives to find slopes, tangent lines, and higher-order derivatives. (36,41,42,56)

d)                  Interpret derivatives in an applied situation. (67,69)

3.3 The Product and Quotient Rules

a)                  Find derivatives of products and quotients. (9,11,17,21,66)

b)                  Compare two ways of taking the derivative. (6,23)

c)                  Find equations of tangent lines. (29)

d)                  Find derivatives using the extended power rule. (32)

e)                  Find derivatives of functions that involve exponentials or a combination of rules. (37,50)

3.4 Derivatives of Trigonometric Functions

a)                  Find limits involving trigonometric functions. (7,10,11)

b)                  Calculate derivatives involving trigonometric functions. (15,17,19,30,33,45)

c)                  Use derivatives of trigonometric functions to find equations of tangent lines. (50)

3.5 Derivatives as Rates of Change

a)                  Relate position, velocity and acceleration.(9,11,15,17,26,28)

b)                  Solve other growth rate applications. (20,35)

3.6 The Chain Rule

a)                  Use Leibniz notation to take the derivative of a composition. (9,12,14)

b)                  Use function notation to take the derivative of a composition. (17,20,28)

c)                  Find the second derivative using the chain rule. (51)

d)                  Solve applications involving the chain rule. (55,57,65)

3.7 Implicit Differentiation

a)                  Find the derivative using implicit differentiation. (8,13,28)

b)                  Find equations of tangent lines using implicit differentiation. (21,26)

c)                  Find derivatives of functions with rational exponents. (33,39)

d)                  Find multiple tangent lines. (49)

3.8 Derivatives of Logarithmic and Exponential Functions

a)                  Find derivatives involving logarithms or b^x. (9,13,15,19,21,38,39,55)

b)                  Find derivatives using the appropriate rule or method. (26,29,57)

c)                  Find tangent lines using logarithmic differentiation. (35)

d)                  Find derivatives using logarithmic differentiation. (44,49)

e)                  Solve applications involving exponential models. (79)

3.9 Derivatives of Inverse Trigonometric Functions

a)                  Find the derivative of functions involving inverse trigonometric functions. (7,9,13,17,24,47)

b)                  Solve applications involving the rate of change of an angle with respect to a side. (29,57)

c)                  Find derivatives of general inverse functions at a given point. (39,52)

3.10 Related Rates

a)                  Solve related rates applications for the rate of change. (8,13,17,19,24,27)

UNIT III

4.1 Maxima and Minima

a)                  Use graphs to illustrate or identify extreme points. (7,8,12,14,17,19,21)

b)                  Use the derivative to locate critical points and extreme points. (27,33,42,49,53,60)

c)                  Solve applications involving extreme points. (45)

4.2 What Derivatives Tell Us

a)                  Sketch functions from properties. (11,43,71)

b)                  Compare f, f ' and f ". (15,65,67,75)

c)                  Determine intervals of increase and decrease. (20,30)

d)                  Use the first derivative test to find extreme points. (34,37,39)

e)                  Determine the concavity on intervals and find inflection points. (47,51,55)

f)                    Use the second derivative test to find extreme points. (61)

4.3 Graphing Functions

a)                  Sketch curves with given properties. (8)

b)                  Sketch functions using analytic methods. (23,31,38)

c)                  Sketch functions with analytic methods and a graphing calculator. (36,63)

4.4 Optimization Problems

a)                  Solve applications by maximizing or minimizing functions. (10,15,16,23,27,31,41)

4.5 Linear Approximation and Differentials

a)                  Write, graph, and use the linear approximation equation. (11)

b)                  Use linear approximations to estimate a quantity. (15)

c)                  Solve applications by estimating the change in a given variable. (23)

d)                  Write the formula for dy for a given function. (32,36)

e)                  Apply Newton's Method to approximate a zero of a function. (see handout)

4.6 Mean Value Theorem

a)                  Find points guaranteed to exist by the Mean Value theorem. (6,16,20)

b)                  Find points guaranteed to exist by Rolle's theorem. (7)

c)                  Solve applications using the Mean Value theorem. (13,29)

4.7 L'Hôpital's Rule

a)                  Evaluate limits of the indeterminate form 0/0. (13,15,23)

b)                  Evaluate limits of the indeterminate form ¥/¥, 0*¥, or ¥-¥. (27,33,37)

c)                  Evaluate limits of the indeterminate form 1^¥, 0^0, or ¥^0. (39,40)

d)                  Compare growth rates. (49,54)

UNIT IV

4.8 Antiderivatives

a)                  Find general antiderivatives and indefinite integrals. (11,17,19,24,26,31,38)

b)                  Find particular antiderivatives and solve initial value problems. (45)

c)                  Relate solutions to initial value problems to their graphs. (49,51,56,61)

d)                  Solve applications involving antiderivatives. (64,65)

5.1 Approximating Areas under Curves

a)                  Understand how Riemann sums represent area under a curve. (1,5)

b)                  Approximate displacement using left, right, or midpoint sums. (9)

c)                  Calculate left, right, and midpoint sums, draw rectangles. (15,19,22)

d)                  Use sigma notation and approximate sums using a calculator. (33,35)

5.2 Definite Integrals

a)                  Understand the properties of the definite integral. (5)

b)                  Approximate net area and definite integrals. (17)

c)                  Relate definite integrals and Riemann sums. (19)

d)                  Relate net area and definite integrals. (26,33)

e)                  Use properties of the definite integral. (39,41,65)

5.3 Fundamental Theorem of Calculus

a)                  Work with area functions. (11,13,21,63,65)

b)                  Evaluate definite integrals using the Fundamental Theorem. (24,25,28,35)

c)                  Find the area of regions bounded by the graph of f. (42,57)

d)                  Find the derivative of functions expressed as integrals. (53)

5.4 Working with Integrals

a)                  Use symmetry to evaluate integrals. (9,11)

b)                  Find the average value of a function. (22,25,27),

c)                  Apply the Mean Value Theorem for Integrals. (34)

5.5 Substitution Rule

a)                  Understand the substitution rule. (6)

b)                  Use the substitution rule to find indefinite integrals. (17,25,27,29,32)

c)                  Use the substitution rule to evaluate definite integrals. (37,39)

6.1 Velocity and Net Change

a)                  Find the displacement over a given interval. (8)

b)                  Determine position functions. (13)

c)                  Determine distance traveled. (17,19)

d)                  Find the position and velocity of an object given the acceleration. (22)

e)                  Find the future value of a quantity given its derivative. (31,33)