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Calculus TI Resources

All of the activities in this FlexBook® resource complement the lessons in our Calculus Student Edition text. Teachers may need to download programs from www.timath.com that will implement or assist with the activities.

What is Calculus?

One-Sided Limits

Create a scatterplot of the data by pressing 2nd[Y =]ENTER and aligning the screen to the right. Create a scatterplot of the data by pressing 2nd[Y =], selecting 1:Plot1, and aligning the screen to the right. Optimizing the volume will help determine the dimension of the box with the largest volume.

Show calculations of the left and right bounds to check if your value that causes the bound exists. One-Sided Bounds www.ck12.org Display left and right bound calculations to verify that your bound value exists.

6. Graph one function at a time by using F 4 to have only one function checked at a time.

Move those Chains

Implicit Differentiation

Using any information you can derive from these statements, make a rule to find the derivative of these functions. One way to find the slope of a tangent drawn to the circle at any point(x,y) located on the curve is by taking the derivative of f1(x) and f2(x). Another way to find the slope of a tangent is by finding the derivative of x2+y2=36 using implicit differentiation.

SE Differentiation - TI Use this result to find the slope of the tangent lines tox2+y2=36 at x=2. To find the derivative of a relation F(x,y), take the derivative of y with respect to x from each side of the relation.

Helicopter Bungee Jump

Optimization

Linear Approximations

In this activity, you will explore relative maxima and minima by drawing a tangent to a curve and observing the slope of the tangent. Press APPS, select the Text Editor application and open extreme1. Observe the slope of the tangent and determine the critical number(s) of the function. When the tangent point is to the left of the relative maximum, the slope of the tangent will be positive, negative, or zero.

What about when the touch point is to the right of the relative maximum. For this function, will the slope of the tangent line be positive, negative, or zero when the point of contact is to the left of the relative minimum. What about when the touch point is to the right of the relative minimum.

Change the window settings so that you no longer see the first 4 seconds of the acceleration-time graph. Linear approximation uses a tangent line to estimate the value of the function near the tangent point. Let the graph on the right be the point where the tangent touches the graph, L(x) is the tangent, and f(x) is the function.

Draw horizontal lines from a, f(x) and the intersection of the vertical line with the tangent. What do you notice about the graph of the function and the graph of the tangent line as you get closer to the tangent point? Now we want to ask the same questions if the tangent point is ata=1.

Graph the line y = 4x + 7. Find the point on the line that is closest to the origin.

Sum Rectangles

FTC Changed History

This activity is intended to complement Calculus, Chapter 4, Lesson 3. Your challenge is to think of at least two ways to estimate the area bounded by the curve=x2 and thex−axis in the interval [0, 1] using rectangles. all rectangles must have the same width. you must construct all the rectangles using the same methods. the base of each rectangle must lie on the axis axis. In the following problem, you will consider three common techniques that use rectangles to find the approximate area under a curve. Remember that the right endpoint is the thex−value and the height is their right endpoint value on the curve.

To summarize it in the calculator, use Home >F3:Calc >4:Sigma for the command with the format:. expression, variable, lower bound, upper bound). Remember that the left endpoint is the thex−value and the height is their left endpoint value on the curve. How would you draw five rectangles, of equal width, so that their midpoints lie on the curve=x2.

Remember that the center point is the x value and the height is the side value of the center point on the curve. Use the Area Approximation program to examine features graphically to complete this part of the activity. What is the smallest value of the integral, and at what value of x is it reached.

9 means the change of sign of the integral from negative to positive that you saw earlier.

Volume by Cross-Sections

Gateway Arc Length

To find the volume, it is only necessary to multiply the area of the pavement by the depth of the pavement. Then use the Numerical Integral command (nInt) in the Calc menu to find the area of the path. You know that the volume of an object is the area of the base times its height.

If the length of one of the sides of this equilateral triangle is 1 cm, calculate the area. If the cross sections perpendicular to the x−axis are equilateral triangles, what is the volume of the solid? If the cross sections perpendicular to the axis are semicircles, what is the volume of the solid.

What is the volume of a solid if all cross-sections perpendicular to their axis are squares? Let the base of the solid be the area of the first quadrant enclosed by the x-axis and one arc of the graph=sin(x). If all sections perpendicular to the x-axis are squares, then approximately what is the volume of the solid.

As dL gets smaller, the difference in the length of dL and the arc length from x to x+dx is eventually infinitesimal. Which of the following integrals gives the length of the graph of y=arcsinx2 betweenx=aandx=b, where 0

The Logarithmic Derivative

Wind tunnel experiments are used to test the wind friction or resistance of a car at different speeds. Remember, to construct a scatterplot, press 2ndY=, select the scatterplot option, and make sure L1 and L2 are selected for Xlist and Ylist. Now draw another scatterplot on your calculator (Plot 2) and graph the new data on the grid above.

After plotting the original wind tunnel data and then plotting the data with the domain and range shifted, what did you notice about the plots of the two sets of points. Find the midpoint between the first points from the first and second scatterplots and the midpoint of the last points from the first and second scatterplots. Thus, ify=ex, then x=ey will be equivalent toy=ln(x) because they are opposites of each other.

6. Graph y = 2x + 3 in a standard viewing window. Use the zero command (2 nd TRACE) to find the x−intercept.

Integration by Parts

Charged Up

What do these integrals have in common that makes them suitable for the substitution method. Use trigonometric identities to rearrange the following integrals and then use the substitution method to integrate. When discharging rapidly, the rate of change of charge,q, with respect to time,t, is directly proportional to the charge.

Show how you solved this differential equation by hand and apply the initial condition (1) =1 to find the specific solution. When you have finished reviewing the set of functions, go to the Y=screen and delete the function.

Infinite Geometric Series

Mr. Taylor, I Presume?

AP Calculus Exam Prep

When the total ratio is greater than 1, explain what happens to the graph and the y-values. If the total ratio is less than -1, describe what happens to the terms of the sequence. The sum of a finite geometric set can be useful for calculating the balance in your bank account, the depreciation of a car, or the population growth of a city.

Use the formulas for the sum of finite and infinite geometric series to complete this problem. Under these conditions, the amount of substances remaining in the body (at 4-hour intervals) forms a geometric series. Complete the table for the amount of the drug in the body in several 4 hour intervals.

Will the patient ever have that much drug in his body if he takes the drug continuously (infinitely) every four hours. Note that the values of the Taylor polynomial and the function values do not match everywhere on the graph. This activity is designed for AP Calculus students as a review for the end-of-year AB Calculus exam.

About how much time you should spend on each question in each part of the exam. AP, College Board, and SAT are registered trademarks of the College Board, which was not involved in the production of and does not endorse this product. To save a value, such as the x−coordinate of the point of intersection, press the [STO→] key.