# Worksheet: Applications of Indefinite Integration

In this worksheet, we will practice using indefinite integration to express a function given its rate of change.

**Q1: **

The slope at the point on the graph of a function is . What is , given that ?

- A
- B
- C
- D

**Q2: **

The area of a lamina is changing at the rate cm^{2}/s,
starting from an area of 60 cm^{2}.
Give an exact expression for the area of the lamina after 30 seconds.

- A
cm
^{2} - B
cm
^{2} - C
cm
^{2} - D
cm
^{2}

**Q3: **

A curve passes through and the tangent at its point has slope . What is the equation of the curve?

- A
- B
- C
- D

**Q4: **

The gradient of the tangent to a curve is . For , the curve has a local minimum value of . Find the equation of the curve.

- A
- B
- C
- D

**Q6: **

A curve passes through the point . The slope of its tangent at a point on the curve is given by . Find the equation of the tangent at the point when is equal to 1.

- A
- B
- C
- D

**Q7: **

The slope at the point on the graph of a function is . Find the equation of the curve if it contains the point .

- A
- B
- C
- D

**Q8: **

The slope at the point on the graph of a function is . Find the equation of the curve if it contains the point .

- A
- B
- C
- D

**Q9: **

The second derivative of a curve is . The curve passes through the point and the gradient of the tangent at this point is . Find the equation of the curve.

- A
- B
- C
- D

**Q10: **

Find the equation of the curve given the gradient of the tangent is and the curve passes through the origin.

- A
- B
- C
- D

**Q11: **

A curve passes through the points and . Find the equation of the curve given the gradient of the tangent to the curve equals .

- A
- B
- C
- D
- E

**Q12: **

The slope at the point on the graph of a function is . What is , given that ?

- A
- B
- C
- D

**Q13: **

Given that the slope at is and , determine .

- A
- B
- C
- D

**Q14: **

A curve passes through and the normal at its point has slope . What is the equation of the curve?

- A
- B
- C
- D

**Q15: **

The slope at the point on the graph of a function is . What is if we know that ?

- A
- B
- C
- D

**Q16: **

Find the equation of the curve given the gradient of the normal to the curve is and the curve passes through the point .

- A
- B
- C
- D
- E

**Q17: **

Find the equation of the curve that passes through the point given that the gradient of the tangent to the curve is .

- A
- B
- C
- D

**Q18: **

Find the local minimum value of a curve given that its gradient is and the local maximum value is 21.

**Q19: **

The gradient of the tangent to a curve is where the value of the local maximum is 9. Find the equation of the curve and the value of the local minimum if it exists.

- A ,
- B , 5
- C ,
- D , 9

**Q20: **

If the rate of change of the sales in a factory is inversely proportional to time in weeks, and the sales of the factory after 2 weeks and 4 weeks are 118 units and 343 units, respectively, determine the sales of the factory after 8 weeks.

**Q21: **

The slope at the point on the graph of a function is . Find the equation of the curve if it contains the point .

- A
- B
- C
- D
- E

**Q22: **

Find the equation of the curve given the slope of the tangent to the curve at its point is and it passes through the point .

- A
- B
- C
- D

**Q23: **

Find the equation of the curve given and the equation of the tangent to the curve at is .

- A
- B
- C
- D

**Q24: **

A curve passes through and the tangent at its point has slope . What is the equation of the curve?

- A
- B
- C
- D

**Q25: **

The rate of change in the area of a metallic plate with respect to time due to heating is given by the relation where the area is in square meters and the time is in minutes. Given that when , find, correct to the nearest two decimal places, the area of the plate just before heating.