### Course Objectives

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:

#### Limits

• 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.

#### Derivatives

• 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.

#### Integrals

• 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.