Calculus Answers

In the temperature range between 0°C and 700°C the resistance R [in ohms] of a certain platinum resistance thermometer is given by

R = 10 + 0.04124T − 1.779 × 10^(−5)T^2

where T is the temperature in degrees Celsius. Where in the interval from 0°C to 700°C is the resistance of the thermometer most sensitive and least sensitive to temperature changes? [Hint: Consider the size of dR/dT in the interval 0 ≤ T ≤ 700.].

Question 03

Write (and draw the graph of) a function which is

(a) Continuous on all points except at x = 1.

(b) Differentiable on all points except at x = 1.

(c) Non-differentiable at five points x = 1, x = 2, x = 3, x = 4 and x = 5.

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]