Gain Mathematical in Calculus through Multiple Representations!

to Learn Calculus! Nancy Norem Powell Bloomington High School Bloomington, IL

Why use these activities?  They help students discover and visualize calculus concepts.  They help students explore and connect calculus concepts.  They emphasize and focus on conceptual understanding of calculus by looking at calculus graphically, numerically, and symbolically.

Derivatives: Activity 1 You are given the graph of a function on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the graph of the derivative of each function over the same interval.

Derivatives: Activity 1 sketch the graph of the derivative of each function over the same interval m=-.8 m=-.7 m=-.5 m=-.3 m=0 m=-.2 m=.2 m=.3 m=.5 m=.7 m=.8

Derivatives: Activity 2 Extrema You are given the graphs of the derivatives of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). For each, indicate the locations of the local maxima and local minima (if any) of the original function.

Derivatives: Activity 2 Extrema: You are given the graphs of the derivatives of f ‘(x). For each, indicate the locations of the local maxima and local minima (if any) of the original function. Positive Derivatives = Increasing Functions Negative Derivatives = Decreasing Functions Local Min Local Min – f ‘ changes from decreasing to increasing and f ” is positive Local Max Local Max– f ’changes from increasing to decreasing and f ” is negative

Derivatives: Activity 3 Mean Value Theorem For each of the function graphs shown, the grid lines are spaced 1 unit apart both horizontally and vertically. Approximate the slope of the secant line connecting the leftmost and rightmost points visible on the graph. If possible, locate at least one point at which the slope of the tangent line is the same as the slope of this secant line.

Derivatives: Activity 3 Mean Value Theorem Approximate the slope of the secant line connecting the leftmost and rightmost points visible on the graph. If possible, locate at least one point at which the slope of the tangent line is the same as the slope of this secant line. Slope is approx. 1/12 Tangent line Secant line

Derivatives: Activity 4 Concavity You are given the graphs of the derivatives y = f ’(x) of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). For each, indicate the locations of the points of inflection for the graphs of the original function y = f (x). Where is the graph concave up or concave down?

Derivatives: Activity 4 Concavity You are given the graphs of the derivatives y = f ’(x). For each, indicate the locations of the points of inflection for the graphs of the original function y = f (x). Where is the graph concave up or concave down? The slope of the first derivative is positive therefore the second derivative is positive and the function is CONCAVE UP CONCAVE DOWN The slope of the first derivative is negative therefore the second derivative is negative and the function is Inflection Point

Derivatives: Activity 5 1. For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 2. For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 3. For all f (x) > 0, f ‘(x) > 0, f “(x) 0, f ‘(x) > 0, f “(x) < 0 4. For all f (x) > 0, f ‘(x) 0 5. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) = 0 6. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) < 0 7. For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 8. For all f (x) 0, f “(x) > 0 9. For all f (x) 0, f “(x) = For all f (x) 0, f “(x) 0, f “(x) < For all f (x) For all f (x) < 0, f ‘(x) < 0, f “(x) = For all f (x) < 0, f ‘(x) < 0, f “(x) < For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy these requirements: Extension: Using the clues on the right and your graphs, make a matching test for your calculus buddy!

Derivatives: Activity 5 - A 1. For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 2. For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 3. For all f (x) > 0, f ‘(x) > 0, f “(x) 0, f ‘(x) > 0, f “(x) < 0 4. For all f (x) > 0, f ‘(x) 0 5. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) = 0 6. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) < 0 7. For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs of f(x) that satisfy these requirements: Which description matches this graph? f(x) has positive y- values The function is increasing, so f‘(x) is positive! The function has no concavity – it is linear so f”(x) = 0! f(x) x b a

Derivatives: Activity 5 - B 1. For all f (x) 0, f “(x) > 0 2. For all f (x) 0, f “(x) = 0 3. For all f (x) 0, f “(x) 0, f “(x) < 0 4. For all f (x) 0 5. For all f (x) < 0, f ‘(x) < 0, f “(x) = 0 6. For all f (x) < 0, f ‘(x) < 0, f “(x) < 0 7. For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy these requirements: Which description matches this graph? f(x) has negative values f (x) is decreasing, so the f ‘(x) is negative ! f (x) is concave down so f “(x) is negative!

Integrals: Activity 6 Area under a curve You are given the graph of a function f on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of each of the following definite integrals by using  x = 1 unit and counting “square units.” If one or more of these definite integrals cannot be estimated for a particular graph, explain why.

Integrals: Activity 6-A Area under a curve You are given the graph of a function f - estimate the value of each of the following definite integrals by using  x = 1 unit and counting “square units.” Interval Approx. Area Total Area sq u sq u -.75 sq u -7.5 sq u.25 sq u

Integrals: Activity 6-B Area under a curve Interval Approx. Area -3 to to to 0 0 to 1 1 to 2 2 to 3 Total Area - 5 sq u sq u sq u sq u = 0 sq u 0 sq u sq u -3.5 sq u

Integrals: Activity 6-C Area under a curve Interval Approx. Area 6 to 5 5 to 4 4 to 3 3 to 2 2 to 1 1 to 0 -1 to 0 -2 to -1 Total Area sq u sq u sq u sq u sq u -.75 sq u -.75 sq u -.25 sq u sq u -.75 sq u -.75 sq u

Integrals: Activity 7 Slope Fields You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the slope field for the antiderivatives of f. Then use this slope field to sketch the graph of an antiderivative F of f over the interval [0,5], assuming F (1) = -2.

Integrals: Activity 7 Slope Fields: a) sketch the slope field for the antiderivatives of f. b) Then use this slope field to sketch the graph of an antiderivative F of f over the interval [0,5], assuming F (1) = -2. f (x) F (x) (0,-1) (3,0) (2,-1/3) (4, 1/3) (5, 2/3) (1,-2/3)xm /3 1/3 -2/3 2/3 0

Integrals: Activity 8 Integrals as Accumulated Area Functions You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the graph of the area function over the interval [1,5] for each function. Then sketch the graphs of the derivative of each area function and compare it with the original function’s graph.

Integrals: Activity 8 Accumulated Area under a curve You are given the graph of a function f - sketch the graph of the area function over the interval [1,5] for the function. Interval Approx. Area Accumulated Area 1 to 1 1 to 2 2 to 3 3 to 4 4 to

Integrals: Activity 9 Trapezoidal rule You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of by using  x = 1 unit and the trapezoidal rule.

Integrals: Activity 9 Trapezoidal rule :estimate the value of by using  x = 1 unit and the trapezoidal rule Trap # Area Approx. Area Note: All of these trapezoids will have a height of 1 unit u u u u u u 2

Integrals: Activity 10 Absolute Value functions and Integrals You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of for each function. (If the graph does not appear over certain parts of the interval, assume that f (x) = 0 for these inputs.)

Integrals: Activity 10 Absolute Value functions and Integrals Estimate the value of for each function. Total Area Vs. Net Area

Integrals: Activity 11 Derivatives and numerical approximation techniques: The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy the given requirements and determine: Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) for will always produce an underestimate?will always produce an underestimate? will always produce an overestimate?will always produce an overestimate? will always produce a exact answer?will always produce a exact answer? will there not be enough information to determine thewill there not be enough information to determine the relationship of the estimation? Tying things together

Integrals: Activity For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 2. For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 3. For all f (x) > 0, f ‘(x) > 0, f “(x) 0, f ‘(x) > 0, f “(x) < 0 4. For all f (x) > 0, f ‘(x) 0 5. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) = 0 6. For all f (x) > 0, f ‘(x) 0, f ‘(x) < 0, f “(x) < 0 7. For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 8. For all f (x) 0, f “(x) > 0 9. For all f (x) 0, f “(x) = For all f (x) 0, f “(x) 0, f “(x) < For all f (x) For all f (x) < 0, f ‘(x) < 0, f “(x) = For all f (x) < 0, f ‘(x) < 0, f “(x) < For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy these requirements: Y-values are positive Function is Increasing Concave up Overestimate given by Right Endpoint Underestimate given by Left endpoint Exact Area given by Unable to determine: Midpoint Simpson’s Trapezoid None