Lernergebnisse:

1. Differentiate functions using implicit differentiation.

2. Differentiate parametric equations with respect to a parameter.

3. Apply implicit and parametric differentiation to solve real-world problems.

Lernergebnisse:

1. Define and identify higher order derivatives of a function.

2. Compute the second and third derivatives of a function.

3. Apply rules of differentiation to find higher order derivatives of polynomial, trigonometric, exponential, and logarithmic functions.

4. Use higher order derivatives to analyze the behavior of functions (e.g., concavity, points of inflection).

Lernergebnisse:

1. Differentiate exponential functions.

2. Differentiate logarithmic functions.

3. Apply logarithmic differentiation to complex expressions.

4. Compute higher-order derivatives of exponential and logarithmic functions.

5. Model and solve real-world problems using exponential and logarithmic differentiation.

Lernergebnisse:

1. Define the concept of related rates and their significance in real-world problems.

2. Use implicit differentiation to solve related rates problems.

3. Set up and solve equations involving related time rates in mathematical and physical contexts.

4. Model and solve life applications involving related time rate problems.

Lernergebnisse:

1. Use the first derivative to determine the intervals where a function is increasing or decreasing.

2. Interpret increasing and decreasing behavior of functions in real-life contexts.

3. Apply calculus techniques to solve problems involving the monotonicity of functions.

Lernergebnisse:

1. Define the concept of a critical point of a function.

2. Identify local maximum and minimum values of a function using the first derivative test.

3. Apply the first derivative test to determine the nature of critical points.

4. Find the maximum and minimum values of a function on a closed interval.

Lernergebnisse:

1. Determine the intervals where a function is concave upward or concave downward.

2. Identify inflection points on the graph of a function.

3. Apply the second derivative test to find local maxima and minima.

4. Sketch curves of functions using information from the first and second derivatives.

Lernergebnisse:

1. Apply the concepts of maxima and minima to solve real-world problems.

2. Use mathematical modeling to find optimal solutions in various contexts.

3. Interpret maximum and minimum values in mathematical and practical situations.

4. Construct mathematical models involving maxima and minima.

Lernergebnisse:

1. Integrate exponential functions.

2. Integrate logarithmic functions.

3. Apply integration techniques to solve geometric problems.

4. Apply integration techniques to solve physical problems.

Lernergebnisse:

1. Find the original function from a given derivative (antiderivatives).

2. Determine the differential of a function.

3. Calculate integrals using the substitution method.

4. Apply the integration by parts technique to evaluate integrals.

Lernergebnisse:

1. Understand the concept of definite integration.

2. Use the fundamental theorem of calculus to compute definite integrals.

3. Apply properties of definite integrals to simplify and evaluate expressions.

Lernergebnisse:

1. Identify area under a curve using definite integration.

2. Calculate the area bounded by a function's curve and the x-axis over a closed interval.

3. Find the volume of a solid generated by revolving a region around the x-axis.