Calculus I Limits and Derivatives

Use linear approximation to estimate the value of \(\sqrt{26}\) by using the function \(f(x) = \sqrt{x}\) at \(a = 25\text{.}\)

The radius of a sphere is measured to be 10 cm with a maximum error of 0.1 cm. Use differentials to estimate the maximum error in the calculated volume (\(V = \frac{4}{3}\pi r^3\)).

A stone is dropped into a still pond, sending out a circular ripple. If the radius increases at a constant rate of 3 cm/s, how fast is the area of the circle increasing when the radius is 10 cm?

The side of a square is measured to be 5 cm with a possible error of 1%. What is the approximate percentage error in the calculated area?

A 13-foot ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 5 feet from the wall?

The edge of a cube is measured with a relative error of 0.02. Use differentials to estimate the relative error in the calculated volume of the cube.

Find the linearization \(L(x)\) of \(f(x) = \sin(x)\) at \(a = 0\text{.}\)

A balloon is being inflated so that its volume increases at a rate of 100 cm³/s. How fast is the radius of the balloon increasing when the diameter is 10 cm?

If \(y = x^4 + 2x\text{,}\) find the differential \(dy\) when \(x = 1\) and \(dx = 0.01\text{.}\)

The demand function for a product is given by \(p = 100 - 0.1x^2\text{.}\) If the production \(x\) is increasing at a rate of 10 units per month, find the rate of change of price \(p\) when \(x = 10\text{.}\)