MA1113: Single Variable Calculus Part 1

1. MA1113: Single Variable Calculus Part 1 Name ________________ Take Home Exercise 3 Part A: Knowledge of Terminology and Symbology ____ Newton's Method ____ Optimization ____ Local Maximum ____ Local Minimum ____ Critical Number ____ Integration ____ Net Area ____ Ledge ____ ____ Inflection Point ____ Change of Variable ____ Second Derivative ____ ____ Slant Asymptote ____ ____ ____ Indeterminate Forms ____ Antiderivative ____ Integration by Parts ____ Fund. Theorem of Calculus A. Any value of x where f "(x)=0 andf '' is changing sign B. The rate of change of the first derivative (measures curvature) C. Name for an infinite sum of infinitesimally small quantities D. A sloping line approached by a function as x approaches  E. Criterion that a function satisfies wherever it is increasing F. Criterion that a function satisfies wherever it is decreasing G. Criterion corresponding to concave down curvature H. Function feature corresponding to f ''(x) > 0 when f '(x)=0 I. Criterion that corresponds to concave up curvature J. Identifies integration and differentiation as inverse processes K. A particular type of inflection point where f '(x) = 0 L. Certain limits that can be evaluated via L'Hospital's Rule M. Function feature associated with an f '(x) change from + to - N. A geometric interpretation of any definite integral O. Terminology that is equivalent to the "indefinite integral" P. Any value of x for which f '(x)=0 or f '(x) is undefined Q. Using the Derivative Product Rule to evaluate some integrals R. Process to find an extremum of a real-world-based function S. Technique that uses the Chain Rule to evaluate some integrals T. Function feature associated with an f '(x) change from - to + U. Uses repeated linearizations to precisely locate a function's root V. Function feature corresponding to f "(x)<0 when f '(x)=0 -1-

2. Part B: Knowledge of Important Formulas 1. L'HOSPITAL'S RULE: a. Complete the formula for L'Hospital's Rule is: b. Indicate at right the indeterminate forms to which the rule may be directly applied (i.e., without any algebraic manipulation): c. After applying the rule once, how do you know if the rule must be applied again? d. Write the proper formula for a second application of L'Hospital's Rule at right: e. How do you know when you no longer need to apply L'Hospital's Rule? 2. RELATIONSHIPS BETWEEN BASIC DIFFERENTIATION AND INTEGRATION FORMULAS -2-

3. 3. INTEGRATION RULES b) Complete the Product Rule: and show how it can be turned into the Integration By Parts Rule: a) If F(x) and G(x) are the antiderivatives of f(x) and g(x) and k is a constant, complete the following rules: Part C: Concept Visualization y 1. NEWTON'S METHOD Derive the basic formula used in Newton's Method to find the root x=r of f(x) using the following outline (which will NOT be supplied on the final exam!) a) Sketch in the linearization L1(x) of f (x) that is tangent to f(x) at x = x1, and write below the formula for the slope of L1: m1 = x b) Show how you can derive the formula for L1(x) using the point-slope formula for a straight line. c) Indicate on the graph the location of the root of L1(x), label it x2, and derive the formula for this root. c) On the sketch indicate how the next linearization root x3 is found and simply write the formula at right: d) At right simply write down the formula for the linearization root xn+1: -3-

4. 2. RIEMANN SUMS (a) 4 a. For the hypothetical function shown in the figures at right, approximate the integral: 3 using a 5-term Riemann Sum with * left-hand sampling points in figure (a), * right-hand sampling points in figure (b). In each case, shade in the area that each Riemann Sum represents. 2 1 5 1 2 3 4 (b) 4 3 2 1 1 2 3 4 5 3. INTEGRATION AS AN AREA COMPUTATION a) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represented by the integral, iii) identify which portions of the total area are positive or negative, iv) give the value of each integral based on your sketches. b) A general rule based on the above illustrations is: "The integral of any function with ______ symmetry over limits that are ___________ with respect to the y-axis is always identically ________." c) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represented by the integral, iii) identify which portions of the total area are positive or negative, iv) give an equivalent integral that is easier to evaluate -4-

5. Part D: Problem Solving 1. Evaluate the following limits using L'Hospital's Rule after suitable algebraic manipulation 2. Use Newton's Method to obtain an estimate for: beginning with a 1st guess of x1=2, and proceeding as follows: a) Show how the above problem can be transformed into finding the root of a polynomial. Provide a rough sketch of the polynomial at right and indicate the location of the root. b) The formula on which Newton's Method is based is given at right. Use this formula as many times as necessary to find the root estimate x3. Perform all your calculations using 5 decimal places. -5-

6. Function Shape Analysis Worksheet Problem: D-3: 1. Find y-intercept and x-intercepts (if readily evident– e.g., f(x) factorable) y-intercept: y = _____ x-intercepts (roots): x = ________________ 2. Determine F(x) Behavior at Infinity (i.e., find Asymptotes, if any exist) Horizontal: y = _________ Slant: y = ___ x Vertical: x = _________ 3. Find Critical Numbers x= c such that Critical Number Summary f ' = 0 at x = __________ f ' DNE at x = ___________ 4. Do 1st Derivative Trend Analysis at Critical Point f '(c)=0 to Identify Local Maxs / Mins / Ledges f ' trend around x = ____: f ' trend is ___ to ___ = ________ f ' trend around x = ____: f ' trend is ___ to ___ = ________ f ' trend around x = ____: f ' trend is ___ to ___ = ________ Critical No. Coords: ( x, y) = _______________ (x,y) = _______________ (x,y) = _______________ Increasing Intervals: ___________________________________________________________ Decreasing Intervals: ___________________________________________________________ BASED ON 1st DERIVATIVE TREND ANALYSIS: At x = ___ , f(x) has a local (max) / (min) / (ledge) At x = ___, f(x) has a local (max) / (min) / (ledge) At x = ___ , f(x) has a local (max) / (min) / (ledge) At x = ___, f(x) has a local (max) / (min) / (ledge) -1-

7. Function Shape Analysis Worksheet (cont) Problem: _____________ f(x) = f'(x)= 5. Find Inflection Points x= c such that and identify intervals of and Infection Point Summary f '' = 0 at x = __________ f '' DNE at x = ___________ Concavity Analysis: i.e., f(x) concave ___ on this interval i.e., f(x) concave ___ on this interval i.e., f(x) concave ___ on this interval i.e., f(x) concave ___ on this interval Inflection Coords: ( x, y) = _______________ (x,y) = ________________ (x,y) = ________________ Concave UP Intervals: ______________________________________ Concave DN Intervals: _____________________________________ 6. Do 2nd Derivative Sign Analysis at each Critical Point: f '(c) = 0 Local Maximums where Local Minimums where to identify and Note 1: If f "(c) = 0 then the 2nd Derivative Sign Analysis is indeterminate. (provides no information) Note 2: Your results here should be consistent your results from the 1st Derivative Trend Analysis on page 1 i.e., concave ____ which is a ______ i.e., concave ____ which is a ______ i.e., concave ____ which is a ______ BASED ON 2nd DERIVATIVE SIGN ANALYSIS: At x = ___ , f(x) has a local (max) / (min) / (ledge) At x = ___, f(x) has a local (max) / (min) / (ledge) At x = ___ , f(x) has a local (max) / (min) / (ledge) At x = ___, f(x) has a local (max) / (min) / (ledge) -2-

8. Function Shape Analysis Results On your sketch of the function be sure to: * draw a symbol at the location of any roots, local maxs or mins, and inflection points * Label the above points with MAX, MIN, or IP as appropriate * identify the intervals over which the function's curvature is concave up (CU) or concave down (CD) 4 3 2 1 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 r 4. A cylindrical drum is required to have a volume of 32π cubic feet. If the material costs of the drum are $2 per square foot for the top and bottom, and $1 per square foot for the side, find the dimensions r and h that will result in the minimum material cost for the drum? h -8-

9. 5. Use the Integration Sum Rule to evaluate: 6. With regard to the integral: a) Make a sketch of the integrand from x = -3 to +3. b) Identify on the sketch the net area represented by the integral. c) Evaluate the integral to find the value of the net area. 7. With regard to the integral: 2 a) Make a sketch of the integrand from x = - to . b) Identify on the sketch the net area represented by the integral. c) Evaluate the integral to find the value of the net area. + - -2 -9-

10. 8. Sketch the integrand from [-π/2 , π/2 ], shade the area being found by the integral and then use Change of Variable technique to estimate that area to three significant digits. 1 1 +/2 + -/2 - -1 -1 9. Use the Substitution Technique (Change of Variable) to evaluate: 10. For the definite integral at right: a) Sketch the integrand from [-π , π ] and shade the area being found. b) Then use the Substitution Technique (Change of Variable) and formula 65 from your Reference Cards to evaluate the integral. -10-

11. 2 2 +/2 +/2 -/2 -/2 -2 -2 11. Use Integration by Parts to evaluate: 12. For the definite integral at right: a) Sketch the integrand and the area being found on the graph. b) Then use Integration By Parts to evaluate the integral. 13. If the integral in problem 3 above had been: (Note the different lower integration limit!) a) Sketch the integrand and the area being found on the graph. b) Determine the value of the integral using the simplest method possible. Explain your reasoning. -11-