Test on How to recognize the graph of the derivative from the graph of a function
How to Recognize the Graph of the Derivative from a Function
Test: Derivatives of Functions
20 questions
Question 1: For a cubic function, its local extrema correspond to the x-intercepts of its derivative graph.
A. Yes
B. No
Explanation: At a local maximum or local minimum of a differentiable function, the tangent line is horizontal, meaning its slope is zero. Consequently, the graph of the derivative will cross the x-axis (f′(x) = 0) at the x-coordinates corresponding to these local extrema. For a cubic function f(x) = ax³ + bx² + cx + d, its derivative f′(x) = 3ax² + 2bx + c is a quadratic function (a parabola). The points where this quadratic derivative crosses the x-axis (i.e., where f′(x) = 0) are precisely the x-coordinates of the local extrema of the original cubic function.
Question 2: The maxima of the sine function correspond to the x-intercepts where its derivative is decreasing.
A. Yes
B. No
Explanation: If f(x) = sin(x), then f′(x) = cos(x). The study materials state that the maxima of sin(x) correspond to the x-intercepts where cos(x) is decreasing. Therefore, the statement is correct.
Question 3: The graph of the function f(x) = eˣ is distinct from the graph of its derivative f′(x).
A. Yes
B. No
Explanation: The study materials state that for the exponential function f(x) = eˣ, its derivative is f′(x) = eˣ. It further clarifies that 'The graph of the function and its derivative are identical,' meaning they are not distinct.
Question 4: A high value on the graph of f'(x) at a point x indicates that the original function f(x) also has a high value at that same point x.
A. Yes
B. No
Explanation: The height of the derivative graph at a point x represents the slope of the original function at x, not the function's value or its height. A high value of f′(x) means f(x) is steeply increasing, not necessarily that f(x) itself is high.
Question 5: For the function f(x) = x³ - 3x, the function f(x) is concave up when x < 0.
A. Yes
B. No
Explanation: For the function f(x) = x³ - 3x, the second derivative is f′′(x) = 6x. According to the study materials, if f′′(x) < 0 (which means 6x < 0, or x < 0), then f(x) is concave down. Therefore, f(x) is concave down when x < 0, not concave up.