Questions: Derivative as Slope of Tangent Line

Question 1

For f(x) = x², the derivative is f'(x) = 2x. What is the slope of the tangent line to the curve at the point (3, 9)?

• A. 3
• B. 9
• C. 6
• D. 2

Answer: (C)

The slope of the tangent line at x = 3 is f'(3) = 2(3) = 6. The value f(3) = 9 is the y-coordinate of the point — it is not the slope. A common error is confusing the output of f with the output of f'.

Question 2

A tangent line to a smooth curve can primarily intersect the curve at the single point of tangency — it cannot cross or touch the curve anywhere else.

• T. True
• F. False

Answer: False

The tangent line is defined by its slope and point of tangency, but as a full line it extends infinitely and can cross the curve at other locations. For example, the tangent to y = x³ at x = 0 is the horizontal line y = 0, which also passes through other points on the curve.

Question 3

What does a horizontal tangent line at a point on a curve tell you about the function at that point?

Model answer: A horizontal tangent means f'(a) = 0, so the instantaneous rate of change is zero at that point — the function is momentarily neither increasing nor decreasing. This often indicates a local maximum, local minimum, or saddle point.

A slope of zero means the function is 'flat' at that instant, but this does not mean the function is constant nearby. f(x) = x³ has a horizontal tangent at x = 0 (f'(0) = 0) yet is still increasing through that point. 'Zero slope at a point' and 'constant near that point' are very different claims.