The Tangent Line Problem

Where calculus actually begins.

12 problems · about 22 minutes · TEKS P.1.A

NameDate

Figure — use these to answer the problems

Move the point and read the tangent slope. On y = x² it is always double the x-coordinate.

y = 1x² + 0x + 0

Point of tangency2

Figure 1

Straightforward practice. Get the method working first.

5 problems

On y = x², what is the tangent slope at x = 6?
Answer
2
Why must a limit be used rather than simply setting h = 0?
- Setting h = 0 gives 0 ÷ 0, which says nothing.
- Because the arithmetic would be harder.
On y = x², tangent slope at x = 3?
Answer
On y = x², tangent slope at x = 0?
Answer
On y = x², tangent slope at x = −4?
Answer

The same ideas with more to keep track of.

3 problems

On y = x², points at x = 2 and x = 4. Secant slope?
Answer
On the line y = 3x + 1, what is the tangent slope anywhere?
Answer
Setting h = 0 in the difference quotient gives which form? Enter 1 for 0 ÷ 0, 2 for a number.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the tangent line process in order.
Write 1 to 4 in the boxes to put these in order.
- Choose a second point a distance h away.
- Write the secant slope as a difference quotient.
- Simplify until h cancels from the denominator.
- Take the limit as h approaches zero.
On y = x², tangent slope at x = 7?
Answer
The Secant
On y = x², the points at x = 1 and x = 3. What is the secant slope?
Answer
The Tangent
On y = x², the tangent slope is 2x. What is it at x = 5?
Answer