On which interval is the average rate of change larger: [1, 2] or [2, 3]?

The midyear population in Japan from 1960 to 2010 has been modeled by the function $P(t)=94,000+\frac{32,658.5}{1+12.75 e^{-0.1706 t}}$ where t is measured in years since 1960 and P(t) is measured in thousands. According to this model, how quickly was the Japanese population growing in 1970? In 1990?

Find equations of the tangent lines to the curve $y=\frac{x-1}{x+1}$ that are parallel to the line x - 2y = 2.

Find numbers a and b such that the given function g is differentiable at 1

$$g(x)=\left\{\begin{array}{ll} a x^{3}-3 x & \text { if } x \leqslant 1 \\ b x^{2}+2 & \text { if } x>1 \end{array}\right.$$

The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t^2 after seconds. What is the maximum height?

Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method to sketch the graph of $f^{\prime}$ below it...

Find h’ in terms of f’ and g’. h(x) = f(g(sin 4x))

The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t^2 after seconds. When does the projectile reach its maximum height?

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by ±0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function

$$B(t)=4.0+0.35sin(2πt/5.4)$$

Find, correct to two decimal places, the rate of increase after one day.

Find the first and second derivatives of the function. \

$$f(x)=\frac{1}{3-x}$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If $f$ is differentiable, then $\frac{d}{d x} \sqrt{f(x)}=\frac{f^{\prime}(x)}{2 \sqrt{f(x)}}$

Find dy/dx by implicit differentiation.

$$x^{3}-x y^{2}+y^{3}=1$$

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maxi - mum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by ±0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function

$$B(t)=4.0+0.35sin(2πt/5.4).$$

Find the rate of change of the brightness after t days.

Differentiate the function. $F(x)=\frac{3}{4} x^{8}$