By the end of Calculus I, students should have begun to build **fundamental knowledge and skills** so they can apply calculus to future STEM academic training and professional practice.

**Fundamental calculus knowledge and skills** will be learned and evaluated based on specific objectives related to:

- Have an intuitive idea of the definition of a limit.
- Evaluate limits (two-sided, left, and right) of the piecewise defined function given algebraically or graphically.
- Calculate infinite limits and detect vertical asymptotes.
- Recognize the precise definition of a limit and demonstrate using it to formally calculate two-sided limits.
- Detect when a function is continuous and when it is discontinuous.
- Apply the Intermediate Value Theorem to mathematically prove two functions intersect on a set interval.

- Apply limits to calculate slopes of tangent lines or instantaneous velocity.
- Given a function, sketch the graph of its derivative and calculate the formula for the derivative.
- Compare the different differentiation formulas and recognize when to use each for given functions.
- Recognize the need for implicit differentiation and apply it to find the slopes of various curves.
- Use differentiation to solve real world problems related to physics.
- Apply implicit differentiation and the chain rule to solve many types of related rates problems.
- Utilize the tangent line or differentials to estimate how a function is changing around a specific point.
- Use the Closed Interval Method to identify absolute maxima and minima of a function.
- State the Mean Value Theorem and identify points on the correct interval that satisfy it.
- Utilize the derivative to determine when a function is increasing or decreasing.
- Use the second derivative to determine when a function is concave up or down.
- Investigate horizontal asymptotes of a function given algebraically by using limits at infinity.
- Summarize all of our current algebra and calculus knowledge to sketch an accurate graph of a function.
- Apply our maxima/minima knowledge to solve optimization problems.
- Recognize how and why Newton’s Method finds intersections between functions.

- Compute general antiderivatives for many types of functions.
- Solve initial value problems for particular antiderivative functions.
- Use antiderivatives to calculate velocity or position from acceleration.
- Estimate the area under a curve using rectangles with heights given by left endpoints or right endpoints.
- Use the limit of finite sums to calculate the definite integral of a function.
- Identify how the definite integral relates with area under the curve.
- Relate slopes and areas through the two parts of the Fundamental Theorem of Calculus.
- Use the antiderivative to calculate definite integrals.
- Calculate the average value of a function over an interval.
- Develop a substitution rule to find antiderivatives of more complicated functions.
- Express the area bounded by two curves as a definite integral and evaluate.
- Identify when it is advantageous to integrate with respect to y instead of x.

- People
- All
- Regular Faculty
- Postdocs
- Visiting Faculty
- Specialists and Instructors
- Adjunct Faculty
- Emeriti
- Graduate Students
- Teaching Assistants
- Staff
- Administration
- Faculty Honors

- Research
- Faculty Research Interests
- Seminars
- Seminars by Week
- Geometry & Topology
- MCIAM
- MathSciNet
- Analysis, PDE, and MathPhys
- Institute of Mathematical Physics
- Math Library
- Phillips Lecture