Calculus Answers

Graph the given​ functions, f and​ g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of

f(x)= -x^3

​g(x)= -x^3-5

Find the area of the triangle formed from the coordinate axes and the tangent line to the curve y = 5x^(−1) −x/5 at the point (5,0).

Question 4

Let f be the function defined by the formula,

f(x) = 1/x+1/x − 10

.

a) Determine the largest possible domain D of f.

b) Is f injective on D?

[8,5]

Question 5

Compute the f ◦ g and its range of the functions f and g below,

f(x) = (x^2 + 5x − 6)(x^2 + 5)/|2x + 3|

, and g(x) = √x + 4

[12]

Question 6

Determine the largest domain, intersection with axes, and sign of f

f(x) = log2(2 −2/x − 3)

[16]

Question 1

Solve the following equation

a) 2e^2x−1 + 5e^x2= 0

b) 2^4x + 2^2x−1 > 2

[8,6]

Question 2

Let f : (−∞, 2] → R be given by

f(x) = √2 − x

a) Show that f is injective.

b) Determine im(f).

c) Find a left inverse g : R → (−∞, 2] of f.

[7,6,5]

Question 3

Let f : Z → Z be given by

f(z) = (2z − 5, if z ≥ 0;

(z + 5, if z < 0.

a) Is f an injective function?

b) Let u ∈ Z, u ≤ 5. show that u ∈ im(f).

c) Let v ∈ Z, v > 5, Show that v ∈ im(f) if and only if v + 5 is even .

[5,6,6]

Question 1

Solve the following equation

a) 2e

2x−1 + 5e

x

2

= 0

b) 2

4x + 22x−1 > 2

[8,6]

Question 2

Let f : (−∞, 2] → R be given by

f(x) = √

2 − x

a) Show that f is injective.

b) Determine im(f).

c) Find a left inverse g : R → (−∞, 2] of f.

[7,6,5]

Question 3

Let f : Z → Z be given by

f(z) = (

2z − 5, if z ≥ 0;

z + 5, if z < 0.

a) Is f an injective function?

b) Let u ∈ Z, u ≤ 5. show that u ∈ im(f).

c) Let v ∈ Z, v > 5, Show that v ∈ im(f) if and only if v + 5 is even .

[5,6,6]

Solve the following I.V.P. by method of Laplace transform:

(i) 𝑦 ′′ + 𝑦 = 𝑓(𝑡), 𝑦(0) = 0, 𝑦 ′ (0) = 0 𝑓(𝑡) = { 2 0 ≤ 𝑡 ≤ 3 3𝑡 − 7 3 < 𝑡 < ∞

(ii) 𝑦 ′′ + 𝑦 = 𝑓(𝑡), 𝑦(0) = 0, 𝑦 ′ (0) = 0 𝑓(𝑡) = { 𝑡^2 0 ≤ 𝑡 ≤ 1 0 1 < 𝑡 < ∞

Can a quadratic function have a range of (-infinity, infinity)?