082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Review exercises 2 113
REVIEWEXERCISES2
1 Statethe ruleand sketch the graphofeach ofthe
followingfunctions:
(a) f(x)=7x−2
(b) f(t)=t 2 −2 0t5
(c) g(x)=3e x +4 0x2
(d) y(t) = (e 2t −1)/2 t 0
(e) f(x)=x 3 +2x+5 −2x2
2 Statethe domainand range ofthe functions in
Question1.
3 Determine the inverse ofeach ofthe following
functions:
(a) y(x) =2x (b) f(t) =8t −3
(c) h(x) = 2x +1 (d)m(r)=1−3r
3
(e) H(s) = 3 s +2
(g) f(t) =e 2t
(i) g(v) = ln(v +1)
4 Iff(t)=e t find
(a) f(2t)
(b) f(x)
(c) f(λ) (d) f(t − λ)
(f) f(v)=lnv
(h) g(v) =lnv+1
(j) y(t) = 3e t−2
5 Ifg(t) =ln(t 2 +1)find(a)g(λ),(b)g(t − λ).
6 Sketchthe following functions:
{ 0 t<0
(a) f(t) =
0.5t t0
⎧
⎨4 t<0
(b) f(t) = t 0t<3
⎩
2t 3t4
(c) f(t)=|e t |
−3t 3
7 Thefunction f (x) isperiodicwith a periodof2, and
f(x)=|x|,−1x1.Sketchffor−3x3.
8 Givena(t) =3t,b(t) =t +3andc(t) =t 2 −3write
expressions
(a) b(c(t))
(c) a(b(t))
(e) a(b(c(t)))
(b) c(b(t))
(d) a(c(t))
(f) c(b(a(t)))
9 Sketchthe following functions,stating any
asymptotes:
(a) y(x) = 3 +x
x
(b) y(x) =
2x
x 2 −1
(c) y(x) =3−e −x (d) y(x) = ex +1
e x
10 Simplify each expression asfar aspossible.
(a) e 2x e 3x
(c) e 4 e 3 e
(e)
( )
e x 2
e −x
(b) e x e 2x e −3x
e x
(d)
e −x
( )
2
(f) ln3x +ln
x
(g) 3lnt+2lnt 2 (h) e lnx
(i) ln(e x ) (j) e lnx2
(k) e 0.5lnx2 (l) ln(e 2x )
(m) e 2lnx (n) ln(e 3 ) +ln(e 2x )
11 Solve the following:
(a) e 4x = 200 (b) e 3x−6 = 150
(c) 9e −x = 54 (d) e (x2) = 60
1
(e) = 0.1
6+e−x 12 Solve the following:
(a) 0.5lnt = 1.2
(b) ln(3t +2) = 1.4
(c) 3ln(t−1)=6
(d) log 10 (t 2 −1) = 1.5
(e) log 10 (lnt) = 0.5
(f) ln(log 10 t) = 0.5
13 Express eachofthe following in terms ofsinhx and
coshx:
(a) 7e x +3e −x (b) 6e x −5e −x
(c) 3ex −2e −x 1
(d)
2 e x +e −x
e x
(e)
1+e x
14 Express eachofthe following in terms ofe x ande −x :
(a) 2sinhx +5coshx
(b) tanhx +sechx
(c) 2coshx − 3 4 sinhx
1
(d)
sinhx −2coshx
(e) (sinhx) 2