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# Calculus I

PREREQUISITE (S): MAT 129 or equivalent or department permission

## COURSE DESCRIPTION:

Introduction to limits, continuity, differentiation, applications of the derivative, the definite and indefinite integral, numerical integration , exponential and logarithmic functions, other transcendental functions, and introduction to differential equations.

## COURSE COMPETENCIES:

During this course, the student will be expected to:

1. Establish the limit of a function.
1.1 Associate the proper limit symbolism with a given graphical situation.
1.2 Calculate limits of certain elementary functions.
1.3 Define the concept of limit for real-valued functions of one real variable .
1.4 Prove that a given limit statement is valid.
1.5 Compute limits involving the trigonometric functions .

2. Determine the continuity of functions.
2.1 State the conditions for the continuity of a function at a point.
2.2 Define continuity on an open interval and on a closed interval.
2.3 Identify intervals of continuity from a given graph.
2.4 Determine points of discontinuity.
2.5 Identify points of discontinuity as removable or non-removable.
2.6 State and apply the Intermediate Value Theorem.

3. Apply the basic rules of differentiation .
3.1 Define the derivative for real-valued functions of one real variable.
3.2 Calculate the derivative of certain elementary functions directly from the definition.
3.3 Calculate derivatives using the appropriate rules for sums, products , and quotients.
3.4 State the connection between differentiability and continuity .
3.5 Calculate higher order derivatives .

4. Differentiate composite functions.
4.1 Calculate derivatives using the chain rule.
4.2 Compute derivatives by the method of implicit differentiation.
4.3 Set up and solve related rate problems.

5. Use the derivative to identify extrema.
5.1 Define relative maximums and minimums of a function.
5.2 Define and find critical values of a function.
5.3 Find the relative extrama of a function using the first and second derivative tests.
5.4 State and apply the Extreme Value Theorem.

6. Identify increasing and decreasing functions.
6.1 Define an increasing (and decreasing) function on an open interval.
6.2 Use the first derivative to determine if a function is increasing (or decreasing) on an interval.
6.3 Determine the open intervals on which a function is increasing and on which it is decreasing.

7. Identify the concavity of a function on an interval.
7.1 Define concave up (and concave down) on an open interval.
7.2 Use the second derivative to determine if a function is concave up (or concave down) on an interval.
7.3 Determine the open intervals on which a function is concave up and on which it is concave down.

8. Find vertical, horizontal and slant asymptotes of a function.
8.1 Define and locate the vertical asymptotes of a function.
8.2 Evaluate infinite limits of a function.
8.3 Use limits at infinity to determine the “end behavior” of a function.
8.4 Use the end behavior of a function to identify any horizontal or slant asymptotes.

9. Apply the derivative to real-world problems.
9.1 Write models for real-world problems.
9.2 Set up and solve applied min/max problems.
9.3 Use the first and second derivative to graph certain elementary functions.
9.4 State the geometrical significance of the first and second derivatives.
9.5 State the physical significance for the first and second derivatives for rectilinear motion.
9.6 State and apply the mean Value Theorem for derivatives.

10. Calculate indefinite and definite integrals.
10.1 Calculate indefinite integrals for elementary functions.
10.2 Calculate Riemann sums in simple cases .
10.3 Define the concept of the definite integral for real-valued functions of one real variable.
10.4 Calculate the definite integral in simple cases directly from the definition.
10.5 State the first and Second Fundamental theorems of calculus.
10.6 Apply the fundamental Theorem of calculus to evaluate definite integrals.
10.7 State the mean Value Theorem for integrals.

11. Find inverse functions.
11.1 Determine whether a function is one to one.
11.2 Define the inverse of a function.
11.3 State the graphical relationship of inverse functions.
11.4 Find the derivative of an inverse function a specified point.

12. Calculate the logarithmic and exponential functions .
12.1 Define the logarithm function in the natural base e.
12.2 Demonstrate the basic properties of logarithms using the definition in 6.1.
12.3 Define logarithms in bases other than e.
12.4 Calculate derivatives and anti-derivatives of the logarithmic functions.
12.5 Define the exponential function in the natural base e.
12.6 Define the exponential functions in based other than e.
12.7 Calculate derivatives and anti-derivatives that are inverse trigonometric functions.

13. Calculate the inverse trigonometric functions.
13.1 Define the inverse trigonometric functions.
13.2 State the domain and range of the inverse trigonometric functions.
13.3 Calculate derivatives of the inverse trigonometric functions.
13.4 Recognize and calculate anti-derivatives that are inverse trigonometric functions.

14. Calculate the hyperbolic trigonometric functions.
14.1 Define the hyperbolic trigonometric functions.
14.2 State the geometrical interpretation of the hyperbolic functions.
14.3 Calculate derivatives and anti-derivatives of the hyperbolic functions.
14.4 Calculate derivatives and anti-derivatives of the inverse hyperbolic functions.

15. Solve simple differential equations.
15.1 Solve differential equations using separation of variables and anti -differentiation.
15.2 Solve differential equations involving exponential growth or decay.

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