1. The rate of change of a function at a point is given by:
Answer:
Explanation:
Derivative provides the instantaneous rate of change of the function.
2. If y = x^2 + 3x, the rate of change of y with respect to x when x = 2 is:
Answer:
Explanation:
dy/dx = 2x + 3. At x = 2, dy/dx = 7.
3. A function has a relative maximum at x = c if:
Answer:
Explanation:
Relative maximum occurs when derivative changes from positive to negative.
4. The tangent to the curve at a point (a, f(a)) is:
Answer:
Explanation:
The equation represents the tangent line using the point-slope form.
5. The derivative of the function gives:
Answer:
Explanation:
Derivative provides the slope as well as conditions for maxima and minima.
6. If a function is increasing in an interval, its derivative in that interval is:
Answer:
Explanation:
An increasing function has a positive slope or derivative.
7. The normal line to a curve at a point is:
Answer:
Explanation:
Normal is always perpendicular to the tangent at the point of contact.
8. The second derivative f''(x) > 0 indicates that the function has a:
Answer:
Explanation:
Positive second derivative indicates the function is concave upward, hence a minimum.
9. For a projectile thrown upwards, the maximum height is achieved when its velocity is:
Answer:
Explanation:
At maximum height, the upward velocity becomes zero before it starts descending.
10. If a company's profit P is given by P(x) where x is the number of items sold, the number of items to be sold to maximize profit is found by:
Answer:
Explanation:
P'(x) = 0 gives critical points and P''(x) > 0 ensures it's a maximum.
11. Approximation of a function value near a point using derivatives is known as:
Answer:
Explanation:
Linear approximation uses the tangent line to approximate function values.
12. The derivative of the distance function with respect to time gives:
Answer:
Explanation:
Rate of change of distance with respect to time is velocity.
13. For a function f(x) = ax^2 + bx + c, if a > 0, the vertex of the parabola represents:
Answer:
Explanation:
If a > 0, the parabola opens upwards and the vertex is a minimum point.
14. The value of the derivative at a point of inflection is:
Answer:
Explanation:
A point of inflection is where the concavity changes, but it doesn't guarantee a zero derivative.
15. The average rate of change of a function over an interval [a, b] is given by:
Answer:
Explanation:
The average rate is the change in function value divided by the change in the independent variable.
16. The process of determining the maximum or minimum values of a function is called:
Answer:
Explanation:
Optimization deals with finding the best values, i.e., maximum or minimum.
17. A function is said to be concave upward in an interval if its:
Answer:
Explanation:
Positive second derivative indicates upward concavity.
18. The slope of the secant line between points (a, f(a)) and (b, f(b)) is:
Answer:
Explanation:
Slope is given by the change in y-values divided by the change in x-values.
19. Rolle's theorem is applicable if:
Answer:
Explanation:
Both conditions must be satisfied for Rolle's theorem.
20. The instantaneous rate of change is given by:
Answer:
Explanation:
Instantaneous rate of change is represented by the derivative.