Homework Answers

The second term of an arithmetic progression is four times the fifth term, and the first term is 10. Find the common difference, and hence the sum of the first 12 terms.

Find the derivatives of each of the following functions:

a. 𝑦 = 5𝑥²+7𝑥−8

b. 𝑓(𝑥) = 7/𝑥⁴

c. 𝑦 = 15/√𝑥

d. 𝑓(𝑥) = 2𝑥^7/2−𝑥^-1/3

e. 𝑦 = (4𝑥²−7𝑥)/𝑥

f. 𝑓(𝑥) = 7𝑥³/√𝑥

Are there any values of p such that p²+48 is equal to -14p?

Differentiate with respect to 𝑥

a. (7𝑥 – 4 )³

b. √(6𝑥+4)

Find the first 3 terms, in ascending powers of 𝑥, of the binomial expansion of (2−𝑥)⁴ and simplify each term.

A grandparent gives a grandchild £100 at birth, and promises to increase the gift by £5 on each subsequent birthday.

a. Show that the grandchild will receive £200 on the 20𝑡ℎ birthday.

b. If the child has saved all the money, what is the total amount at age 20?

c. By how much would the gift have to increase each year if the total at age 20 is to be £4,200?

a. Expand (𝑎+𝑏)⁵. Hence find the coefficient of 𝑥 in the expansion of (4𝑥+2/9𝑥)⁵

b. The coefficient of 𝑥² in the expansion of (1+𝑥)ⁿ is 45. Given that 𝑛 is a positive integer, find the value of 𝑛.

The curve 𝑦=−𝑥³+3𝑥²+6𝑥−8 cuts the 𝑥-axis at 𝑥=−2,𝑥=1 and 𝑥=4.

a. Sketch the curve, showing clearly the intersection with the coordinate axes.

b. Differentiate 𝑦=−𝑥³+3𝑥²+6𝑥−8

c. Show that the tangents to the curve at 𝑥=−2 and 𝑥=4 are parallel.