Calculus – Differentiation on Differentials and Small Increments

1. Suppose the earth were a perfect sphere and we determined its radius to be 3959 ± 0.1 miles. What effect would the tolerance of ± 0.1 have on our estimate of the earth’s surface area?

2. About how accurately should we measure the radius of a sphere to calculate the surface area within 1% of its true value?

3. Estimate the allowable percentage error in measuring the diameter of a sphere if the volume is to be calculated correctly to within 3%.

4. (Measuring variations in g) The period P of a pendulum of length L, is

given by P=2 π √ Lg , where g is the acceleration of gravity.

a. Assuming that L remains fixed, show that a 1% increase in g results in approximately a 0.5% decrease in the period.

b. For fixed g, what percentage change in L will produce a 1% in P?5. The length s of a side of a box in the shape of a cube is measured at 2

feet and the volume (V = s3) is then estimated to be 8 ft2.a. Estimate the error in the volume calculation of the measurement

of s is inaccurate by ± 2 feet.b. If the volume maximum allowable error in the measurement of s

in inches, if the volume calculation is to have an error of at most 1ft2.

6. The top circular cover of a trampoline is to be increased, its radius going from 2 m to 2.3 m. What will the approximate change in the area of the top be?

7. Find the approximate value of 1

0.205.

8. A ball radius 7 cm has to be altered to a radius of 6.5 cm. What will the volume of the smaller ball be?

9. A pane of glass is being thickened by 0.3 cm. on a sheet of glass 1.5 m by 0.7 m, with thickness 1.2 cm what will be the approximate change in the volume?

10.What is the approximate value of √171?11.A cylindrical glass tumbler will have its height increased from 10.5 cm

to 11.2 cm. If the radius of 3 cm is to remain constant, what will be the approximate volume of the new tumbler?

12.A wooden barrel needs to have the radius decreased by 0.5 cm. The barrel had radius 25 cm and height 1.1 m. If the height remains constant, find the approximate change in the volume.

13.The volume of a cubic container will be decreased by 1.5%. Find the percentage change in the length of the sides.

14.A metal sphere has a percentage error of ± 2.1% in the volume. Find the percentage error in the radius.

15.The volume of a cone has to be decreased by 1.7%. Find the percentage decrease in the radius if the height remains constant.

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Calculus – Differentiation on Differentials and Small Increments

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Calculus – Differentiation on Differentials and Small Increments

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