Mathematics I · Unit 1 · 5 hrs
Function of One Variable
Exam-focused notes for Function of One Variable (Mathematics I, MTH117): what the TU syllabus asks and how it has actually been tested, with 11 solved past questions from this unit.
What this unit covers
- Four ways of representing a function
- Linear mathematical model
- Polynomial, Rational, Trigonometric, Exponential and Logarithmic functions
- Combination of functions
- Range and domain of functions and their Graphs
Range and domain of functions and their Graphs
Sketch the graph of $f(x) = x^2$. Find its domain and range.
Evaluate $\lim_{x \to 1^-} \sin^{-1} \left( \frac{1-\sqrt{x}}{1-x} \right)$ [5+5]
- Function: $f(x) = x^2$ - Limit to evaluate: $\displaystyle \lim{x \to 1^-} \sin^{-1}\left(\frac{1-\sqrt{x}}{1-x}\right)$ - Marks: [5 + 5] --- $x^2$ is defined for every real number (no division or even-root restrictions). $$\text{Domain} = (-\infty, \inft...
Full solved answer →Sketch the graph and find the domain and range of the function $f(x) = 2x - 1$. [5]
$$f(x) = 2x - 1, \quad \text{i.e., } y = 2x - 1$$ This is a linear function (degree 1 polynomial) with slope $m = 2$ and y-intercept $-1$. --- A polynomial function has no restrictions (no division by zero, no even roots, no logs). It is defined for every r...
Full solved answer →Find the domain and sketch the graph of the function $f(x) = x^2 - 6x$
Estimate the area between the curve $y = x^2$ and the line y = 1 and y = 2. [3+2]
(a) Domain and Sketch of $f(x) = x^2 - 6x$ - Function: $f(x) = x^2 - 6x$ $f(x) = x^2 - 6x$ is a polynomial, defined for all real $x$. $$\text{Domain} = (-\infty, +\infty) = \mathbb{R}$$ Intercepts: - $x$-intercepts: $x^2 - 6x = 0 \Rightarrow x(x-6) = 0 \Rig...
Full solved answer →Linear mathematical model
As dry air moves upward, it expands and cools. If the ground temperature is $20°C$ and the temperature at height of 1 km is $10°C$, express the temperature $T$ (in $°C$) as a function of the height $h$ (in kilometer), assuming that linear model is appropriate.
(a) Draw a graph of the function in part (b). What does the slope represent?
(c) What is the temperature at a height of 2.5 km?
[5+5]
- Ground level: $h = 0$ km, $T = 20^{\circ}C$ → point $(0, 20)$ - At $h = 1$ km: $T = 10^{\circ}C$ → point $(1, 10)$ - Linear model assumed: $T = mh + b$ - Required: temperature at $h = 2.5$ km Slope: $$m = \frac{10 - 20}{1 - 0} = \frac{-10}{1} = -10$$ Inte...
Full solved answer →Dry air is moving upward. If the ground temperature is $20^\circ$ and the temperature at a height of 2km is $10^\circ$, express the temperature $T$ in $^\circ$C as a function of the height $h$ (in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km? [5]
- At ground level: $h = 0$ km, $T = 20^\circ$C → point $(0, 20)$ - At height $h = 2$ km, $T = 10^\circ$C → point $(2, 10)$ - Linear model assumed. --- Linear form: $T = mh + b$. Slope: $$m = \frac{T2 - T1}{h2 - h1} = \frac{10 - 20}{2 - 0} = \frac{-10}{2} = ...
Full solved answer →Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function $T = 0.03t + 8.50$, where T is temperature in degree centigrade and t represents years since 1900. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. [5]
Linear model: $$T = 0.03t + 8.50$$ - $T$ = temperature in degrees Celsius - $t$ = years since 1900 --- The equation has the form $T = mt + c$ with slope $m = 0.03$ and intercept $c = 8.50$. Slope ($m = 0.03$): Represents the rate of change of temperature wi...
Full solved answer →Questions
Question 1: If $f(x) = x^2$ then find $\frac{f(2+h)-f(2)}{h}$
Question 2: Dry air is moving upward. If the ground temperature is $20°C$ and the temperature at a height of $1$ km is $10°C$, express the temperature $T$ in $°C$ as a function of the height $h$ (in kilometers), assuming that a linear model is appropriate.
(b) Draw the graph of the function in part (a). What does the slope represent?
(c) What is the temperature at a height of $2$ km?
Question 3: Find the equation of the tangent to the parabola $y = x^2 + x + 1$ at $(0, 1)$. [2.5+5+5]
Part 1: $f(x) = x^2$; evaluate the difference quotient at $x = 2$. Part 2: Dry air moving upward. - Ground level: $h = 0$ km, $T = 20^\circ C$ - Height: $h = 1$ km, $T = 10^\circ C$ - Linear model assumed. Part 3: Parabola $y = x^2 + x + 1$; tangent require...
Full solved answer →Combination of functions
Question
If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$, then find $f \circ g$ and its domain and range.
A rectangular storage container with an open top has a volume of $20 \text{ m}^3$. The length of its base is twice its width. Material for the base costs Rs 10 per square meter; material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base.
[5+5]
(a) Finding fog, its Domain and Range $$f(x) = \sqrt{x}, \qquad g(x) = \sqrt{3-x}$$ $$fog(x) = f(g(x)) = f\left(\sqrt{3-x}\right) = \sqrt{\sqrt{3-x}} = (3-x)^{1/4}$$ $$\boxed{fog(x) = (3-x)^{1/4}}$$ Domain analysis via composition: - $g(x) = \sqrt{3-x}$ req...
Full solved answer →If $f(x) = x^2 - 1$, $g(x) = 2x + 1$, find $fog$ and $gof$ and domain of $fog$. [5]
$$f(x) = x^2 - 1$$ $$g(x) = 2x + 1$$ Required: $fog$, $gof$, and domain of $fog$. --- $$fog(x) = f(g(x)) = f(2x+1)$$ Replace $x$ in $f(x)=x^2-1$ with $(2x+1)$: $$fog(x) = (2x+1)^2 - 1$$ $$= 4x^2 + 4x + 1 - 1$$ $$\boxed{fog(x) = 4x^2 + 4x}$$ --- $$gof(x) = g...
Full solved answer →If $f(x) = \sqrt{2-x}$ and $g(x) = \sqrt{x}$, find $f \circ f$ and $f \circ g$. [5]
$$f(x) = \sqrt{2-x}, \qquad g(x) = \sqrt{x}$$ Required: $f \circ f$ and $f \circ g$. --- $$(f \circ f)(x) = f(f(x)) = \sqrt{2 - f(x)} = \sqrt{2 - \sqrt{2-x}}$$ Domain: Inner root requires: $$2 - x \geq 0 \implies x \leq 2$$ Outer root requires: $$2 - \sqrt{...
Full solved answer →If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$, find $g \circ f$ and $f \circ g$. [5]
- $f(x) = \sqrt{x}$ - $g(x) = \sqrt{3-x}$ Required: $g\circ f$ and $f\circ g$ (with domains). All data present and readable. --- $$(g\circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{3 - \sqrt{x}}$$ Domain: For $f(x)=\sqrt{x}$: need $x \geq 0$. For $\sqrt{3-\sqrt...
Full solved answer →Make Unit 1 stick
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