# h, Limits, and the Derivative

### Objective

The definition of a derivative involves the limit (as *h* approaches zero) of the quotient of (*f*(*x* + *h*) − *f*(*x*)) divided by *h*.

In this informal activity you will use a sketch which allows you to manipulate such a quotient, and observe the behavior as you change the value of *h*.

### Steps:

Step 1: Open the sketch called **Derivati.gsp** in the *Derivat* folder. You can change the function by dragging the roots left or right, and by dragging the *a* slider.

Step 2: Double-click the "Show x and f(x)" action button. Move the point *x* left and right on the axis, and watch the value of *f*(*x*) change.

Step 3: Double-click the next two buttons to show (*x* + *h*) and *f*(*x* + *h*). Then drag the green slider back and forth to observe how you can control the value of *h* and make the point *(x + h)* on the *x*-axis approach point *x*. Notice that you can also double-click the "Animate h" button to change *h* automatically.

Step 4: Double-click the button to show the secant and its slope. The secant line connects the points that represent*f*(*x*) and *f*(*x* + *h*). Then drag or animate *h* and watch how the secant line changes as *h* changes.

Step 5: Double-click the button to show the tangent line and its slope.

Q1. Does the tangent line depend on the value of *h*?

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Q2. What is the relationship between the secant and tangent lines as *h* approaches zero?

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Step 6: Use Sketchpad's calculator to compute the value of (*f*(*x* + *h*) − *f*(*x*))/*h*. (To use the calculator, choose **Calculate** from the Measure menu. You can then click on the calculator keys, and on measurements in the sketch, to form the expression you want to calculate.)

Q3. What measurement in the sketch corresponds to this calculation?

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Q4. What measurement corresponds to the limit of this calculation as *h* approaches zero?

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### Explore More:

Open the other sketchesin the *Derivat* folder and explore how they work. What conjectures can you make about each of these sketches?