Calculus Answers

Allocated function: y = A cos kt

Values assigned: A = 4, k = 5

Graph of displacement y (metres) against time t (seconds) for time interval from t = 0s to t = 2π/k seconds.

Q: Calculate the turning points of the function using differential calculus and show for each turning point whether it is a maximum, a minimum or a point of inflexion, by using the second derivative of the function.

The coefficient of x in the Maclaurin's expansion of sinx sink is

what is the area under the curve y(x)= 3e^-5x from x = 1 to x= ∞

If 𝑅 𝑢 = 𝑢 − 𝑢 2 𝑖 + 2𝑢 3 𝑗 − 3𝑘, find �𝑑� �� �� ׬ .�� 𝑏. න 1 2 𝑅(𝑢)𝑑�

If 𝐴 = 3𝑥 2 + 6𝑦 𝑖 − 14𝑦𝑧𝑗 + 20𝑥𝑧 2𝑘, ��׬ ��evaluate 𝐴 ∙ 𝑑𝑟 from (0,0,0) to (1,1,1) along the following path C: 𝑎. 𝑥 = 𝑡, 𝑦 = 𝑡 2 , 𝑧 = 𝑡 3 b. the straight lines from (0,0,0) to (1,0,0) and then to (1,1,0) and the to (1, 1, 1) c. the straight line joining (0,0,0) to (1,1,1).

The acceleration of a particle at any time t ≥ 0 is given by 𝑎 = 𝑑𝑣 𝑑𝑡 = 12 cos 2𝑡 𝑑𝑡𝑖 − 8 sin 2𝑡 𝑗 + 16 𝑡 𝑘 , if the velocity v and displacement r are zero at t=0, find 𝑣 𝑎𝑛𝑑 𝑟 at any time.

Given that P = (-4, 5) and Q = (-6, 4), find the component form and magnitude of the vector PQ.

Explain Fundamental Theorem of Calculus (Integrate a variety of functions)

State the chain rule, the product rule and the quotient rule of differentiation. For each rule, give an example of a trigonometric expression which can be differentiated using the rule.

How to complete the rational numbers? Mention the processes.