## 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.