MAT 210 - Calculus I

Outcome 2: Calculate the derivative of a function and apply the derivative to solve application problems.
    Task 1:    Employ the connection between the slope of the secant line to the slope of the tangent line.
    Task 2:    Use the definition of the derivative to evaluate the derivative of a function.         
    Task 3:    Use various notations for the derivative such as y’, dy/dx, f’(x), d/dx(y), etc.          
    Task 4:    Employ the connection between continuity and differentiability.
    Task 5:    Use the definition of the derivative to calculate one-sided derivatives.
    Task 6:    Evaluate derivatives of constants using the Constant Rule.
    Task 7:    Evaluate derivatives of functions of the form y = x^n, n any rational number, by using the Power Rule.
    Task 8:    Use the Constant Multiple Rule to calculate derivatives.
    Task 9:    Use the Sum and Difference Rule to calculate derivatives of the sum or difference of functions.
    Task 10: Calculate derivatives of products of functions by using the Product Rule.
    Task 11: Use the Quotient Rule to calculate the quotient of two functions.
    Task 12: Evaluate derivatives of compositions of two or more functions using the Chain Rule.
    Task 13: Evaluate higher-order derivatives.
    Task 14: Calculate y’ for an implicitly defined function defined by f(x,y) = 0.
    Task 15: Use the definition of the derivative to calculate the derivatives of sine and cosine.
    Task 16: Use the differentiation rules to calculate the derivatives of tangent, cotangent, secant, and cosecant.
    Task 17: Use the Mean Value Theorem to locate the value(s) of x where the average rate of change of a function over the interval [a,b] is equal to the instantaneous rate of change.

Outcome 3: Employ the relationship between the derivative, antiderivative, indefinite integral, and definite integral.
    Task 1:    Determine the critical points of a function.
    Task 2:    Determine the intervals where a function is increasing or decreasing.
    Task 3:    Determine the inflection points of a function.
    Task 4:    Determine the intervals where a function is concave up or concave down.
    Task 5:    Use the First Derivative Test to determine local extrema.
    Task 6:    Use the Second Derivative Test to determine local extrema.
     Task 7:    Distinguish between local and absolute extrema.
    Task 8:    Sketch a curve using the information obtained in Task 1-Task 7 of Outcome 3.
    Task 9:    Use the derivative to solve a maximum/minimum application problem.
    Task 10: Use the derivative to solve a related-rates type problem.
    Task 11: Approximate the zeros of a function using Newton’s Method and a graphing utility.
    Task 12: Use the derivative to calculate the velocity, speed, and acceleration of a particle moving along a curve.
    Task 13: Develop a linear approximation to a function y = f(x) about x = c.
    Task 14: Use differentials to approximate the exact change in a function.
    Task 15: Use the position function s(t) = ½gt^2 + v_0 t + s_0 to model a free-falling object.

Outcome 4: Calculate the integral of a function to find the antiderivative and the area under a curve.
    Task  1: Demonstrate that the general antiderivative is not unique.
    Task  2: Calculate the general antiderivative for polynomials, radical functions, and basic trigonometric functions.
    Task  3: Relate finding antiderivatives to differential equations and initial-value problems.
    Task  4: Use antiderivatives to solve initial-value problems involving velocity/acceleration, free-falling bodies, and related rates.
    Task  5: Relate the definite integral to the antiderivative.
    Task  6: Relate the indefinite integral of a function to its general antiderivative.
    Task  7: Partition an interval [a,b] into n closed subintervals of equal length.
    Task  8: Determine the width, delta x, of each subinterval.
    Task  9: Determine the height of the rectangle by using the left endpoint, right endpoint, or midpoint of the subintervals.
    Task 10: Calculate the area of each rectangle.
    Task 11: Estimate the area under the curve y = f(x) by adding the areas of all the rectangles.
    Task 12: Write the sums for the areas using summation notation.
    Task 13: Employ algebraic rules for finite sums.
    Task 14: Use the formulas for finite sums to calculate various summations.
    Task 15: Use a graphing utility to calculate approximations to areas under a curve y = f(x).
    Task 16: Use numerical methods to calculate definite integrals.
    Task 17: Calculate the Riemann sum for y = f(x) on the interval [a,b].
    Task 18: Use a graphing utility to evaluate the Riemann sum of a function y = f(x).
    Task 19: Demonstrate that the definite integral is defined to be the limit of the Riemann sum.
    Task 20: Use the integral symbol to represent the definite integral of y = f(x) over the interval [a,b].
    Task 21: Discuss terminology concerning integration.
    Task 22: State the condition that insures which functions are continuous.
    Task 23: Demonstrate the rules for definite integrals.
    Task 24: Utilize the Mean Value Theorem for the definite integral.
    Task 25: Use the Fundamental Theorem of Calculus, Parts I and II, to calculate the definite integral.
    Task 26: Use graphing utility to calculate the definite integral for a function.
    Task 27: Use the notation of the indefinite integral.
    Task 28: Discuss the need for the constant of integration on an indefinite integral.
    Task 29: Calculate the indefinite integral for various functions.
    Task 30: Use u-substitution to integrate functions which are made up of a composition of two or more functions.
    Task 31: Use the method of u-substitution to solve both definite and indefinite integrals.
    Task 32: Discuss how one might develop a numerical method for calculating definite integrals.
    Task 33: Use the Trapezoid Rule to calculate definite integrals.
    Task 34: Use Simpson’s Rule to calculate definite integrals.
    Task 35: Discuss error involved in using either the Trapezoid Rule or Simpson’s Rule.
    Task 36: Use a graphing utility to integrate a function using the Trapezoid Rule or Simpson’s Rule.

Outcome 5: Employ exponential, logarithmic, and inverse trigonometric functions, utilizing derivatives and integrals.
    Task 1:    Calculate the natural log function.
     Task 2:    Graph and interpret the function y = e^x.
     Task 3:    Solve exponential and logarithmic equations.
    Task 4: Differentiate exponential functions.
     Task 5:    Integrate exponential functions.
    Task 6: Set up and solve applications involving exponential functions.
     Task 7:    Graph and interpret the function y = ln x.
     Task 8:    Use the properties of logs to simplify log expressions.
     Task 9:    Differentiate natural log functions.
     Task 10: Perform logarithmic differentiation.
     Task 11: Integrate functions resulting in the natural log function.
     Task 12: Identify the domain and range of the inverse trig functions.
     Task 13: Use right triangle interpretation of inverse trig functions to find associated values.
     Task 14: Differentiate inverse trig functions.
     Task 15: Integrate functions resulting in inverse trig functions.