# Worksheet: Applications of Indefinite Integration

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

**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
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- 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 .

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- B
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- E

**Q12: **

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

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**Q13: **

Given that the slope at is and , determine .

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**Q14: **

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

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- D

**Q15: **

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

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**Q16: **

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

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**Q17: **

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

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**Q18: **

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

- ALocal minimum value is 34.5.
- BLocal minimum value is 48.
- CLocal minimum value is 4.5.
- DLocal minimum value is .

**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 .

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- 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 .

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- D

**Q23: **

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

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**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.