Differentiating standard functions

Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, B, and the number of wildflower plants, F, at time t weeks after the start of the study.

In the model B = 20 + 2t + cos 3t and F = 50e^0.1t.

The model assumes that B and F can be treated as continuous variables.

Suggest a reason why this model may not be valid for values of t greater than 12.

In this question you must show detailed reasoning.

Solve the following equations.

f ′ (x) = 0

f ′′ (x) = 0

The diagram shows the graph of the gradient function y = f ′ (x) for the domain 0 ⩽ x ⩽ 2 π.

Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points A, B, C and D.

Explain how to use the graph of the gradient function to find the values of x for which

f(x) is increasing.

Using set notation, write down the set of values of x for which f(x) is increasing in the

domain 0 ⩽ x ⩽ 2 π.

Substance A is decaying exponentially such that its mass is m grams at time t minutes. Find the missing values of m and t in the following table.

Substance B is also decaying exponentially, according to the model m = 160e^-0.055t, where m grams is its mass after t minutes.

Determine the value of t for which the mass of substance B is half of its original mass.

Determine the rate of decay of substance B when t = 15.

State whether substance A or substance B is decaying at a faster rate, giving a reason for your

answer.

A particle moves in the x - y plane so that at time t seconds, where t ⩾ 0, its coordinates are given by

x = e^2t - 4e^t + 3, y = 2e^-3t.

Explain why the path of the particle never crosses the x-axis.

Determine the exact values of t when the path of the particle intersects the y-axis.

Hence find the coordinates of the particle when its path is parallel to the y-axis.

Differentiate the following with respect to x.

Differentiate the following with respect to x.

(2x + 3)⁷

x³ ln x

Find the equation of the curve through (1, 3) for which = 6x - 5.

By considering show that the curve is convex for all values of x.

In this question you must show detailed reasoning.

Find the x-coordinate of the point where the curve crosses the x axis.

The points A and B lie on the curve and have x coordinates 2 and 4. Show that the line AB is parallel to

the x-axis.

Find the coordinates of the turning point on the curve.

Determine whether this turning point is a maximum or a minimum.