## MTH 131 Chapter Objectives

### Chapter 2

1. State and explain the informal definition of limit.
2. Estimate limits graphically and numerically.
3. State and explain, with examples, the ways limits may fail to exist.
4. Estimate and evaluate one-sided limits.
5. Use one-sided limits to justify that a limit does not exist.
6. Determine one-sided and two-sided infinite limits.
7. Find and sketch the vertical asymptotes of the graph of a function.
8. Use limit laws to evaluate limits.
9. Use substitution to evaluate limits.
10. Use algebraic techniques to resolve 0/0 indeterminate forms.
11. Use the squeeze theorem to evaluate limits.
12. Use trigonometric techniques to resolve 0/0 indeterminate forms.
13. Use the definition of continuity to determine if a function is continuous at a point.
14. Use the properties of continuity to determine if a function is continuous at a point or on an interval.
15. Classify discontinuities.
16. Apply the Intermediate Value Theorem.
17. Use the formal definition of limit to prove certain limits.
18. Find a delta corresponding to a particular epsilon by using analytical, numerical, or graphical techniques.

### Chapter 3

1. State, explain, and illustrate the limit definition of derivative.
2. Use the limit definition of derivative to evaluate a derivative.
3. Interpret the derivative as the slope of a tangent line.
4. Find equations of tangent lines.
5. Determine when a derivative does not exist.
6. Graph the derivative function from the graph of a given function.
7. Evaluate derivatives (and higher-order derivatives) using basic differentiation rules.
8. Calculate an average rate of change (i.e., slope of a secant line).
9. Interpret the derivative as an instantaneous rate of change.
10. Solve problems involving motion along a line.
11. Solve various application problems involving rates of change.
12. Evaluate derivatives of trigonometric functions.
13. Identify compositions of functions, and write a function as a composition of two functions.
14. Use the chain rule to differentiate compositions of functions.
15. Use implicit differentiation to find the derivative of an implicitly-defined function.
16. Find equations of the lines tangent and normal to the graph of an implicitly-defined function.
17. Compute the derivative of an inverse function.
18. Evaluate derivatives involving the inverse trigonometric functions.
19. Compute the derivative of an exponential function of any base.
20. Compute the derivative of a logarithmic function of any base.
21. Use logarithmic differentiation.

### Chapter 4

1. Use implicit differentiation to relate rates.
2. Solve application problems involving related rates.
3. Determine the linearization of a function at a point.
4. Use the linearization at a point to approximate function values near the point.
5. Compute differentials.
6. Find the critical numbers of a function.
7. Determine whether an extreme value is absolute or relative.
8. Find the absolute extreme values of a continuous function on a closed, bounded interval.
9. State, explain, and apply Rolle's Theorem.
10. State, explain, and apply the Mean Value Theorem.
11. Use the first derivative to find intervals on which a function is increasing/decreasing.
12. Use the first derivative test to locate relative extrema.
13. Use the second derivative to find intervals on which the graph of a function is concave up/down.
14. Find the points of inflection of the graph of a function.
15. Apply the second derivative test to classify relative extrema.
16. Evaluate limits at infinity.
17. Find the horizontal asymptotes of the graph of a function.
18. Use calculus techniques to solve application problems involving optimization.
19. Identify indeterminate forms.
20. Apply L'Hopital's rule to resolve indeterminate forms.
21. Use algebraic techniques to rewrite limits so that L'Hopital's rule applies.
22. Use Newton's method to approximate the zeros of a function.
23. Use tangent lines to illustrate Newton's method.
24. Use basic integration rules to evaluate indefinite integrals.
25. Solve initial value problems.

### Chapter 5

1. Use rectangles to approximate the area of a bounded region under the graph of a function.
2. Use upper and lower sums to approximate area.
3. Compute a Riemann sum for a function on an interval.
4. Define and interpret the definite integral of a function on an interval.
5. Use Riemann sums to approximate definite integrals.
6. Use properties of the definite integral to simplify and evaluate integrals.
7. Use area to evaluate definite integrals.
8. Use the definite integral to define the average value of a function.
9. Use the Fundamental Theorem of Calculus to evaluate definite integrals.
10. Use and evaluate definite integrals in applications involving area and average value.
11. Interpret an integral with variable bounds as an accumulation function.
12. Use the Fundamental Theorem of Calculus to differentiate functions defined by integrals.
13. Use substitution to evaluate indefinite integrals.
14. Use substitution to evaluate definite integrals.
15. Evaluate integrals involving exponential functions.
16. Evaluate integrals that result in logarithmic functions.
17. Evaluate integrals that result in inverse trigonometric functions.

Last updated August 17, 2020