Basic Mathematics · Unit 5
Applications of Derivatives
Exam-focused notes for Applications of Derivatives (Basic Mathematics, MTH104): what the TU syllabus asks and how it has actually been tested, with 13 solved past questions from this unit.
What this unit covers
- Tangent and normal lines to curves
- Average and instantaneous rates of change
- Related rates problems
- Optimization and extrema
- Absolute maximum and minimum values
- Local extrema and critical points
- First derivative test
- Second derivative test
- Concavity and inflection points
- Curve sketching
- Rolle's theorem
- Mean value theorem
- L'Hospital's rule
Mean value theorem
Verify mean value theorem for the function $f(x) = x^2 + 3x + 1$ in $[-1,1]$. [5]
- Function: $f(x) = x^2 + 3x + 1$ - Interval: $[a, b] = [-1, 1]$, so $a = -1$, $b = 1$ If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ - $f(x) = x^2 + 3x + 1$ is...
Full solved answer →State Mean value theorem. Verify the mean value theorem if $f(x) = x^2 + 2x - 1$ on $[0, 1]$. [1+4]
- Function: $f(x) = x^2 + 2x - 1$ - Interval: $[0, 1]$, so $a = 0$, $b = 1$ All data present and readable. If a function $f(x)$ is: - Continuous on the closed interval $[a, b]$, and - Differentiable on the open interval $(a, b)$, then there exists at least ...
Full solved answer →Rolle's theorem
State Rolle's Theorem and show that $x^3 + 3x + 1 = 0$ has exactly one real solution. Find the area of the region enclosed by the parabola $y = 2 - x^2$ and the line y = -x. [5+5]
- Function/equation: $x^3 + 3x + 1 = 0$ If a function $f$ is: 1. continuous on the closed interval $[a, b]$, 2. differentiable on the open interval $(a, b)$, and 3. $f(a) = f(b)$, then there exists at least one point $c \in (a, b)$ such that $f'(c) = 0$. Le...
Full solved answer →Concavity and inflection points
Determine the concavity of y = 3 + sin x on [0, 2π\piπ]. [5]
- Function: $y = 3 + \sin x$ - Interval: $[0, 2\pi]$ $$y' = \cos x$$ $$y'' = -\sin x$$ Concave up requires $y'' 0$: $$-\sin x 0 \implies \sin x < 0$$ On $[0, 2\pi]$, $\sin x < 0$ for $x \in (\pi, 2\pi)$. Concave down requires $y'' < 0$: $$-\sin x < 0 \impli...
Full solved answer →Define the concavity of the function. The graph of the function is then $f(x) = x^4 - 4x^3 + 10$. Find the intervals on which $f$ is increasing and on which $f$ is decreasing. Find where the graph of $f$ is concave up and where it is concave down. Find the local maximum or local minimum value of function if exist. [2+3+3+2]
- Function: $f(x) = x^4 - 4x^3 + 10$ --- A function $f$ is concave up on an interval if its graph lies above all its tangent lines on that interval (opens upward, shape $\cup$). Equivalently, $f''(x) 0$. A function $f$ is concave down on an interval if its ...
Full solved answer →Determine the concavity and find the inflection point of the function $f(x) = x^3 - 3x^2 + 2$. [5]
- Function: $f(x) = x^3 - 3x^2 + 2$ $$f'(x) = 3x^2 - 6x$$ $$f''(x) = 6x - 6$$ Set $f''(x) = 0$: $$6x - 6 = 0 \implies x = 1$$ - For $x < 1$ (test $x = 0$): $f''(0) = 6(0) - 6 = -6 < 0$ → concave down on $(-\infty, 1)$ - For $x 1$ (test $x = 2$): $f''(2) = 6...
Full solved answer →Optimization and extrema
Define Newton's-Raphson method with their formula.An open top box is to be made by cutting small congruent squares from the corners of square sheet of tin having length 12 inch. and is bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?[2+8]
Definition: The Newton-Raphson method is an iterative numerical technique for finding the roots (zeros) of a real-valued function $f(x) = 0$. Starting from an initial guess $x0$, it uses the tangent line to the curve to generate successively better approxim...
Full solved answer →Tangent and normal lines to curves
Find the equations of tangent and normal to the curve $x^3 + y^3 - 9xy = 0$ at the point $(2, 4)$. [5]
- Curve: $x^3 + y^3 - 9xy = 0$ - Point: $(2, 4)$ $$2^3 + 4^3 - 9(2)(4) = 8 + 64 - 72 = 0 \checkmark$$ $$3x^2 + 3y^2\frac{dy}{dx} - 9\left(y + x\frac{dy}{dx}\right) = 0$$ $$\frac{dy}{dx}(3y^2 - 9x) = 9y - 3x^2$$ $$\frac{dy}{dx} = \frac{9y - 3x^2}{3y^2 - 9x} ...
Full solved answer →L'Hospital's rule
What is L'Hospital's rule? Using this rule evaluate the following: [1+4]
L'Hospital's Rule:
L'Hospital's rule states that if $\lim_{x \to a} \frac{f(x)}{g(x)}$ produces an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
provided the limit on the right exists (or is infinite).
Evaluate:
-
$\lim_{x \to 0} (\sec x)^{\frac{1}{x^2}}$
-
$\lim_{x \to 0} \frac{x - \sin x}{x^3}$
- Limit 1: $\displaystyle \lim{x \to 0} (\sec x)^{1/x^2}$ - Limit 2: $\displaystyle \lim{x \to 0} \frac{x - \sin x}{x^3}$ - Marks split: [1 + 4] All required data present. --- If $f$ and $g$ are differentiable near $x = a$ (with $g'(x) \neq 0$ near $a$) and...
Full solved answer →Local extrema and critical points
If $f(x) = x^2 + 2x - 1$ and $g(x) = 2x - 3$, then find $fog(x)$ and $gof(x)$. Find the local maxima and local minima of the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$. [5+5]
- $f(x) = x^2 + 2x - 1$ - $g(x) = 2x - 3$ - $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ (for Part 2) --- $$fog(x) = f(g(x)) = f(2x-3)$$ Substitute $(2x-3)$ into $f$: $$= (2x-3)^2 + 2(2x-3) - 1$$ $$= (4x^2 - 12x + 9) + (4x - 6) - 1$$ $$= 4x^2 - 8x + 2$$ $$gof(x) = g(f(...
Full solved answer →Average and instantaneous rates of change
A rock breaks loose from the top of a tall cliff. Find average speed during the first 2 sec of fall. What is its average speed during the 1sec interval between second 1 and second 2? Find the speed of the falling rock at $t = 1$ and $t = 2$. [3+3+4]
- Rock breaks loose from rest, so initial velocity $u = 0$ - Free fall under gravity: $g = 9.8 \text{ m/s}^2$ (standard value) - Equations of motion: $s = ut + \frac{1}{2}gt^2$, $v = u + gt$ (Note: This is a standard Halliday/Resnick style problem where $g ...
Full solved answer →Related rates problems
Water runs into a conical tank at the rate $9\text{ ft}^3/\text{minutes}$. The tank stands point down and has a height of $10\text{ ft}$ and a base radius of $5\text{ ft}$. How fast is the water level rising when the water is $6\text{ ft}$ deep? [5]
- Inflow rate: $\dfrac{dV}{dt} = 9 \text{ ft}^3/\text{min}$ - Cone, point down: height $H = 10$ ft, base radius $R = 5$ ft - Find: $\dfrac{dh}{dt}$ when $h = 6$ ft Step 1: Volume of a cone $$V = \frac{1}{3}\pi r^2 h$$ Step 2: Similar triangles $$\frac{r}{h}...
Full solved answer →Absolute maximum and minimum values
Find the absolute maximum and minimum values of $f(x) = x^{2/3}$ on the interval $[-2, 3]$. [5]
- Function: $f(x) = x^{2/3}$ - Interval: $[-2, 3]$ (closed) $$f'(x) = \frac{2}{3}x^{-1/3} = \frac{2}{3\sqrt[3]{x}}$$ Set $f'(x) = 0$: The numerator is $2 \neq 0$, so $f'(x)$ is never zero. Where $f'(x)$ is undefined: $\sqrt[3]{x} = 0 \Rightarrow x = 0$. Sin...
Full solved answer →Make Unit 5 stick
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