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Math 1210-003 Camacho Spring 2012 Midterm Review Applications of the Derivative Critical Points: There are three types of critical points... 1. Stationary points are the values of x where (e.g. numerator is 0). 2. Singular points are values of x where 3. End points are the boundary points of the domain on which (e.g. on the interval the end points are and ) is undefined (e.g. denominator is 0). is being considered. Extreme Values: Maximum and Minimum values of a function 1. The extreme values of a function MUST occur at a critical point 2. To find the extreme values of a function first find the critical points, then evaluate the function at all known critical points to determine which is the maximum value and which is the minimum value. Monotonicity & Concavity: 1. 2. 3. 4. 5. means means means means means is increasing is decreasing has a horizontal tangent line is concave up is concave down Optimization Problems: 1. Determine all quantities of interest (e.g. h = height, w = width, A = area) 2. Determine the relationships between quantities (e.g. 3. Identify the objective function (e.g. 4. Use the relationships to write the objective function in terms of only one variable (e.g. , therefore ). , ) ) Math 1210-003 Camacho 5. Spring 2012 Midterm Review Determine the maximum/minimum value of the objective function by first finding all critical points and then plugging the critical points into the objective function to determine which yields the optimal value. In some problems you are looking for the maximum value and in others you will be looking for the minimum value. Newton’s Method To solve , start with an initial guess, . Then take successive guesses according to the following recursive formula until your approximation is reasonably close to the true solution: Integration Sigma Notation Rules 1. 2. 3. Special Formulas 1. 2. 3. Math 1210-003 Camacho Spring 2012 Midterm Review Indefinite Integrals (Anti-Differentiation): 1. 2. 3. 4. 5. 6. 7. 8. Fundamental Theorems of Calculus: 1. 1st Fundamental Theorem: 2. 2nd Fundamental Theorem: Method of Substitution (Indefinite Integrals): 1. 2. Identify inner function and set equal to u. (e.g. for the inner function Determine du by using the formula (e.g. if then so ). ) 3. Substitute u and du into the integral so that it is completely in terms of u. (e.g. ) 4. Anti-differentiate in terms of u. (e.g. 5. Substitute x into the solution using the definition of u. (e.g. ) ) Math 1210-003 Camacho Spring 2012 Midterm Review Method of Substitution (Definite Integrals): 1. Find the anti-derivative, , using the u-substitution. 2. Evaluate the integral by calculating Calculating Exact Anti-derivatives: 1. 2. Given a function's derivative, , and some known value for the function, , you should be able to use this information to determine . a. Use the techniques of integration to anti-differentiate of including the " + C ". b. Use the fact that and find to determine C. Given a function's second derivative, , some known value for the function's first derivative , and some known value for the function , you should be able to use this information to determine . a. Use the techniques of integration to anti-differentiate of including the " + C ". b. Use the fact that c. Use the techniques of integration to anti-differentiate of including the " + D ". d. Use the fact that and find to determine C. to determine D and find Math 1210-003 Camacho Spring 2012 Midterm Review Practice Problems Section 3.1: #5 - 26 Section 3.2: #1 - 10 Section 3.2: #11 - 18 Section 3.3: #6 - 10 Section 3.4: #10 - 16 Section 3.7: #1 – 16 Section 3.8: #1 - 36 (finding critical points and extreme values) (finding where a function increases or decreases) (finding where a function concave up or concave down) (finding relative maxima and minima) (optimization problems) (Newton’s Method) (finding anti-derivatives) Section 4.1: #15 – 24 Section 4.3: #9 – 16 Section 4.4: #1 - 14 Section 4.4: #15 - 26 Section 4.4: #35 - 52 (sigma notation) (properties of definite integrals) (evaluating definite integrals) (finding anti-derivatives using the substitution method) (evaluating definite integrals using the substitution method)