# Worksheet: Finding Functions given Their Derivatives

Q1:

Find the function of the curve whose first derivative is and the function equals 7 when equals .

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Q2:

Determine the function if and .

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Q3:

Determine the function if , , and .

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Q4:

Determine the function satisfying , , and .

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Q5:

Find the equation of the curve through the point with the property that the slope of the normal to the curve at is .

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Q6:

If 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 metres, 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.

Q7:

A group of labourers are digging a hole, where the rate of change of the volume of the sand removed in cubic metres with respect to the time in hours is given by the relation . Calculate the volume of the sand dug out in 5 hours rounded to the nearest hundredth.

Q8:

Suppose that and when . Find in terms of .

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Q9:

Determine if .

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Q10:

Given that , and when , find the relation between and .

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Q11:

Determine the function if , , and .

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Q12:

Find the function on which satisfies , , and .

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Q13:

A function is such that, at all points, the product of the slope of its graph and the square of its -coordinate is 3. Find the equation of the curve given the curve passes through the point .

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Q14:

The rate of change of the volume of a gas, in cubic metres, with respect to its pressure is given by , where is a constant. Find the relationship between the volume of the gas and the pressure when and .

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Q15:

Find the equation of the curve passing though the point given the gradient of the tangent at any point is . Then find the tangents at the points on the curve which intersect with the line .

• Athe curve: , the tangents: and
• Bthe curve: , the tangents: and
• Cthe curve: , the tangents: and
• Dthe curve: , the tangents: and

Q16:

The slope at each point of a curve is inversely proportional to . The slope is 8 when and . Determine in terms of .

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Q17:

Determine the function , if , when , , and .

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Q18:

The function satisfies for some constants and . If the graph of has an inflection at and a local minimum value at , what is ? If the graph also has a local maximum, identify it.

• A , local maximum value:
• B , local maximum value:
• C , local maximum value:
• D , local maximum value:

Q19:

The equation of the tangent to a curve at is . Find the equation of the curve given .

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Q20:

The rate of change of the gradient to the curve that passes through the points and is . Find the equation of the curve.

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Q21:

The graph of passes through and the slope of the tangent at is . What is ?

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Q22:

The function satisfies the relation where and is a constant. Find given and .

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Q23:

Determine the function such that , , and .

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Q24:

The second derivative of a function is and the equation of the tangent to its graph at is . Find the equation of the curve.

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Q25:

Determine the function such that , and .

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