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Calculus TI Resources
This FlexBook® resource contains Texas Instruments (TI) resources for the TI-89 unless otherwise noted in the description of the activity. There are also corresponding links in the student edition of the CK-12 Calculus FlexBook® textbook, http://www.c k12.org/book/CK-12-Calculus/. Any activity that requires a calculator file or program, go to http://www.education.ti.com/calculators/downloads and type the name of the activity or program in the search field.
What is Calculus?
One Sided Limits
Before changing the value of a, students will graphically estimate the limit of y1(x) as x approaches 1 from the left and right. Before changing the value of a, students will graphically estimate the limit of y2(x) as x approaches 1 from the left and right. Before changing the value of a, students will graphically estimate the limit of y3(x) as x approaches 2 from the left and right.
Move Those Chains
Implicit Differentiation
Students are first asked to make a guess about the derivative of f(x) = (2x+1)2 based on the power rule. Students are asked to make a guess about what the derivative of f(x) = (2x+1)2 is using the power rule. To access theDeriv command, go to the Math menu (2. [MATH]) and select B:Calculus >A:nDerivat(.) Students only need to graph the functions y2 and y3.
Helicopter Bungee Jump
Optimization
Linear Approximation
In this activity, students investigate relative maxima and minima by drawing tangents to a curve and making observations about the slope of the tangent. Students are asked whether the slope of the tangent to the left of a relative maximum is positive, negative, or zero. Students are asked whether the slope of the tangent to the left of a relative minimum is positive, negative, or zero.
Students are asked whether the slope of the tangent line to the right of a relative minimum is positive, negative, or zero. For the questions at the end, students are expected to be familiar with the concept of the definite integral. To test for understanding of calculus concepts, questions that connect a graph of the derivative to the original function are effective.
Students often justify increasing a position by saying, “this one is going up.” They should avoid using 'it'. For example: "the derivative is positive, so the function is increasing.". To find the exact coordinates of the point, students take the first derivative (Menu > Calculation > Derivative), find the critical value (Menu > Algebra > Solve), and take the second derivative. As you get closer to the tangent point, the graph of the function and the graph of the tangent line appear to be the same.
Have them use the slope and the point (-1, 4) to get the equation of the line.
Sum Rectangles
FTC Changed History
Students should know how to write an arithmetic sequence that will be part of the summation expression they will use to find approximate areas. To find the approximation of the area under the curve, students must use the formula given on the worksheet and then also find the sum of the last column. To account for the width of the interval, they must have the∗0.2 at the end of the summation.
Students must draw rectangles on their graph using midpoints and then complete the table. The area of each rectangle is 0.02 * the height of the rectangle. Discuss with the students whether they think this is an overestimation or an underestimation of the true area. In addition, students should answer that the concavity of the function and whether it is increasing or decreasing will determine which estimates are overestimated versus which are underestimated.
For example, when the function is decreasing and concave down, the function curves steeper for the second half of the rectangle than the first. Fory=x2, an increasingly concave upward function in this domain, approaches the center 7123 from the left and the trapezoid method approaches the value of the definite integral from the right. Students are led to the brink of a discovery of a discovery of the Fundamental Theorem of Calculus, which dxd.
This investigation must follow the definition of the definite integral and the relationship between the integral of a function and the area of the region bounded by the graph of the function and the x-axis.
Volume by Cross Sections
Gateway Arc Length
Students should already be familiar with the concept of the integral and use it to find the area under a curve. Students must find the volume of a new trajectory using the graph and calculator. They will use the Pythagorean Theorem to approximate the solution and use Calculus to find the exact solution.
They will also use CAS capabilities, including arcLen( ), to solve a variety of arc length questions. The syntax for arcLenisarcLen(f(x),x,a,b) where f(x) is the function, x the variable and the arc length from x=a to x=b is found. This activity will help students approximate arc length and use calculus to find the exact arc length.
Law of cosines c2=a2+b2−2abcosθwhere θ is the angle between a and b. The Pythagorean theorem is a special case of this where θ=90◦. For Exercise 2, students should use CAS to find the arc length of the Gateway Arch equation. Arc length for parametric equations is presented and students must solve this arc length by hand.
Students are expected to know the arc length formula and answer multiple choice questions without a calculator.
The Logarithmic Derivative
Then they will draw a line and use the x and y intercepts to create the graph of the inverse. They are asked to comment on their observations of the two scatterplots on the worksheet. Students are instructed to find the midpoint between the first point on each of the scatterplots and the midpoint between the last points for each of the scatterplots on the graph.
Then, to find the equation of the line, they must find the slope using the two midpoints and the point-slope form. They can find the equation of the line connecting these two points by finding the slope and then using the slope-intercept form. Switch domain and range, switch x, and reflect the graph of the function over liney=x in the equation.
Students will determine the derivative of the function y=ln(x) and work with the derivative of both y=ln(u) and y=loga(u). In the process, the students will show thatlimh→0ln(a+h) )−ln(a). In this problem, students are asked to use thelimitcommand (F3:Calc >3:limit() to find the values. In this section, students are asked to use the derivative command (F3:Calc >1:d(differentiate) to find the derivative.
Students are asked to identify u(x) and a for each function and then find the derivative by hand or use the Derivative command to find the derivative.
Integration by Parts
Charged Up
Opportunities are provided for skill development and practice of the method of taking integrals of suitable functions. Begin by reviewing the method of differentiation of composite functions (the "chain rule") and the methods of integration of the standard function forms. Make sure students are comfortable with this and then challenge them to consider more difficult forms – in this case, compound functions of the form= f(g(x)) which may be suitable for integration by substitution methods.
Then they integrate it with respect and rewrite the evaluation of the integral back into the original variable. Note: Some students may not realize that they need to have the 12 before the integral. This research provides opportunities for revision and consolidation of previous key concepts related to standard integrals and the product rule for differentiation, as well as mastery of the method of integration by parts.
Students should be given the opportunity to review and strengthen their skills and understanding of the product rule and the process of integration (especially distinguishing between the integral of a function and the area under the curve of the graph of that function. A graphical approach can help students appreciate the importance of each component of the parts of the integration statement by by parts: Areas under a curve by considering and respecting are key concepts here Integration by parts allows students to find the integral of a function that is a product by taking the derivative of one component and the integral of the other.
To view the graph of the function family, students must type clGDB diffq1 on the HOME screen.
Infinite Geometric Series
Mr. Taylor, I Presume?
AP* Calculus Exam Prep
They will consider the effect of the value for the common ratio and the first term using the TRANSFRM APP for the TI-84. An extension would be to solve infinite geometric series that converge using sigma notation and the limit of the partial sum formula. Students will now change the equation in the Transfrm application to explore changes in the graph of the general series, an=a1·rn−1.
On the worksheet, we show students the derivation of the formula for the sum of a finite geometric set. Students can use the formula to find the sum of the series given on the worksheet. They will consider the effect of the value of the common ratio and determine whether the infinite geometric series converges or diverges.
They will also consider the derivation of the sum of a convergent infinite geometric series and use it in solving several problems. At the conclusion of the introductory problem, a Taylor polynomial form centered at zero is found. At this time, put the general form of the Taylor polynomial when x=0 on the board.
As students compare the values of the Taylor polynomial to the values of the original function, they will notice that the values are closest to the center (dex − value where the derivatives are found) and become further apart as the thex − values are further away. the middle (the graph shows this further).
