Math 105A/B Test One Name:
1. (5 marks) Find dy
dx for 5 of the following equations.
a. y= ln(πsin(ex))
c. y= sin(2x)
e. y=xe2x
g. y=π3
i. y= 3esin(2x)
b. y=(log10 x)3
tan x
d. esin y=x
f. y= (2 + cos x)2
h. x3+y3=9
2xy
j. y= arctan(2x)
1
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main points of this exam paper are: Arctan, Limits, Antiderivative, Functions, Inverse Function, Domains, Ranges, Graphs, Solution, Problems
Typology: Exams
2012/2013
1 / 7
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Math 105A/B Test One Name:
- (5 marks) Find dydx for 5 of the following equations. a. y = ln(π sin(ex)) c. y = sin(2x) e. y = xe^2 x g. y = π^3 i. y = 3esin(2x)
b. y = (log tan^10 xx)^3 d. esin^ y^ = x f. y = (2 + cos x)^2 h. x^3 + y^3 = 92 xy j. y = arctan(2x)
- (5 marks) Evaluate 5 of the following limits a. limx→e ln x^ x−−e^1 c. limh→ 0 2 cos(5π/ h4+h)+√^2 e. limx→∞ (^) x^32 g. limx→∞ (^2) xx 2 i. limx→∞ (^1) x
b. limx→∞ e−x^ ln x d. limx→ 0 sin(arccos(2 2 x)) f. limx→ 0 5 x^3 − 10 x − 2 h. limx→ 0 1 −cos(4 5 x 2 x) j. limx→∞ e^3 x^22 x+^3 x
- (10 marks) Find the inverse function (including their domains and ranges) for the following and sketch their graphs. a. f (x) = 2x + 5, −∞ < x < ∞. y
x
c. f (x) = ex, −∞ < x < ∞. y
x
b. f (x) = sin x, −π 2 ≤ x ≤ π 2. y
x
d. f (x) = 10, −∞ < x < ∞. y
x
- (10 marks) Solve 2 of the following problems. a. Find the solution to the IVP: f ′′(x) = −8, f ′(0) = 1 and f (0) = 0.
b. Find the solution to the IVP: f ′(x) = 5f (x) and f (0) = 100.
c. Find the solution to the IVP: f ′′(x) = − 4 f (x) and f (0) = 6, f ′(0) = 2.
b. (10 marks) Where does f (x) = ex^ − 5(x − 1)^2 achieve a maximum value on 12 ≤ x ≤ 72? Show all your work.