Applications of Derivatives: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
In the following graph estimate the open intervals over which the function is increasing or decreasing.
$y=\frac{x^{3}}{4}-3x$
Apply the first derivative criterium.

Problem 2
Look at the graph of the function $f(x)=x^{4}-2x^{2}$
What can you say about the following propositions involving its graph?
(i) It is increasing on $(-1,1)$
(ii) It is decreasing on $(-\infty ,-1)$
(iii) It is increasing on $(-1,0)$
(iv) It is decreasing on $(0,1)$
Problem 3
Identify the open intervals over which the function $h(x)=27x-x^{3}$ is increasing or decreasing.
Problem 4
Find the derivative of the function $y=x+\frac{4}{x}$. Determine when the function increases and decreases.
Problem 5
Given the function $f(x)=\left( x-1\right)^{2}\left( x+3\right)$

a) The critical points of $f$ are $\left( 1,0\right) ;\left( -\frac{5}{3},\frac{256}{27}\right)$

b) The function is increasing on $\left( -\infty ,-\frac{5}{3}\right) \cup \left( 1,\infty \right)$ and decreasing on $\left( -\frac{5}{3},1\right)$

c) The function has a maximun at $\left( -\frac{5}{3},\frac{256}{27}\right)$ and has a minimum at $\left(1,0\right)$
Problem 6
Let $f(x)$ be defined as $f(x)=x^{4}-32x+4$

a) The critical point of $f$ is at $x=4$ only.
b) The function is increasing on $\left(-\infty, 2\right)$
c) The function has a minimum at $x=2$
Problem 7
Graph the function $f(x)=\left( x+2\right)^{2/3}$

a) The critical point of $f$ is $(0,0)$ only.
b) The function is increasing on $\left( -\infty ,-2\right)$ and and decreasing on $(-2,\infty )$
c) The function has a maximun at $\left(-2,0\right)$
Problem 8
Find the inflection points of the function $f(x)=\frac{1}{4}x^{4}-2x^{2}$ and analyze the concavity of the function

A) Concave: $\left( \frac{2\sqrt{3}}{3},\infty \right)$, inflection point: $\left( \frac{2\sqrt{3}}{3},-\frac{20}{9}\right)$

B) Convex: $\left( \frac{2\sqrt{3}}{3},\infty \right)$, inflection point: $\left( \frac{2\sqrt{3}}{3},-\frac{20}{9}\right)$

C) Convex: $\left( -\infty ,-\frac{2\sqrt{3}}{3}\right) \cup \left( \frac{2\sqrt{3}}{3},\infty \right)$, concave on: $\left( -\frac{2\sqrt{3}}{3},\frac{2\sqrt{3}}{3}\right)$, inflection points: $\left( -\frac{2\sqrt{3}}{3},-\frac{20}{9}\right) ;\left( \frac{2\sqrt{3}}{3},-\frac{20}{9}\right)$

D) Concave: $\left( -\infty ,-\frac{2\sqrt{3}}{3}\right) \cup \left( \frac{2\sqrt{3}}{3},\infty \right)$, convex on: $\left( -\frac{2\sqrt{3}}{3},\frac{2\sqrt{3}}{3}\right)$, inflection points: $\left( -\frac{2\sqrt{3}}{3},-\frac{20}{9}\right) ;\left( \frac{2\sqrt{3}}{3},-\frac{20}{9}\right)$
Problem 9
Consider the following function $f(x)=2x^{4}-8x+3$
Find the points of inflection and analyze the concavity.

A) Convex on $\left( -\infty ,\infty\right)$ and there are no points of inflection.

B) Convex on $\left( -\infty ,0\right)$, concave on $\left( 0,\infty \right)$, a points of inflection at $\left( 0,3\right)$

C) Concave on $\left( -\infty ,0\right)$, convex on $\left( 0,\infty \right)$, a point of inflection at $\left( 0,3\right)$ D) Concave on $\left( -\infty ,\infty \right)$, there are no points of inflection.
Problem 10
Determine the concavity of $y=-x^{3}+3x^{2}-2$

A) $f$ is concave on $\left( -\infty,1\right)$, convex on $\left( 1,\infty \right)$

B) $f$ is convex on $\left( -\infty,1\right)$, concave on $\left( 1,\infty \right)$

C) $f$ is concave on $\left( -\infty,0\right)$, convex on $\left( 0,\infty \right)$

D) $f$ is convex on $\left( -\infty,2\right)$, concave on $\left( 2,\infty \right)$

Problem 11
Determine the concavity of $f(x)=-x^{3}+6x^{2}-9x-1$

A) $f$ is concave on $\left( -\infty,1\right)$, convex on $\left( 1,\infty \right)$
B) $f$ is concave on $\left( -\infty,2\right)$, convex on $\left( 2,\infty \right)$
C) $f$ is convex on $\left( -\infty,0\right)$, concave on $\left( 0,\infty \right)$
D) $f$ is convex on $\left( -\infty,2\right)$, concave on $\left( 2,\infty \right)$
Problem 12
Let we have the function $f(x)=x^{4}-4x^{3}+2$
1. Find all relative extremes and inflection points.
2. Use the criterion of the second derivative where appropriate.

A) Minimum at $(3,-25), f$ is convex on $\left( -\infty ,0\right) \cup \left( 2,\infty \right)$, concave on $\left(0,2\right)$
B) Maximum at $(3,-25), f$ is convex on $\left( -\infty ,0\right)$, concave on $\left( 2,\infty \right)$
C) Maximum at $(0,2), f$ is convex on $\left( 0,2\right)$, concave on $\left( 2,\infty \right)$
D) minimum at $(0,2), f$ is convex on $\left( 0,2\right)$, concave on $\left( 2,\infty \right)$
Problem 13
Find all relative extreme and inflection points of the function: $f(x)=x^{2/3}-3$.

A) Maximum at $\left( 0,-3\right);$ concave on $\left( -\infty, 0\right)$
B) Maximum at $\left( 0,-3\right);$ concave on $\left( 0,\infty\right)$
C) Minimum at $\left( 0,-3\right);$ concave on $\left( -\infty,0\right) \cup \left( 0,\infty \right)$
D) Maximum at $\left( 0,-3\right);$ convex on $\left( -\infty,0\right) \cup \left( 0,\infty \right)$
Problem 14
Let $f(x)=\left( \left( x^{2}+3\right)^{5}+x\right)^{2}$.
Find $\frac{d}{dx}\ f(-1)$
Problem 15
Let $f(x)=\sqrt{2+\sqrt{2+\sqrt{x}}}$
Find $\frac{d}{dx}\ f(4)$
Problem 16
Find the equation of the tangent line to the function $f(x)=\sqrt{25-x^{2}}$ at point $(3,4)$

A) $4y+3x=25$ is the tangent line.
B) $4x+3y=25$ is the tangent line.
C) $3y-4x=25$ is the tangent line.
D) None of the above.
Problem 17
The displacement of its equilibrium position for an object in harmonic motion located at the end of a spring is:
$y=\frac{1}{3}\cos 12t-\frac{1}{4}\sin12t$ where $y$ is measured in feet and $t$ in seconds.

Determine the position and speed of the object when $t=\frac{\pi}{8}$.

A) Position: $\frac{1}{4}$feet, speed: $4$ ft/sec
B) Position: $0$ feet, speed: $2$ ft/sec
C) Position: -$\frac{1}{4}$feet, speed: $4$ ft/sec
D) Position: $0$ feet, speed: $-2$ ft/sec
Problem 18
A 15 cm pendulum moves according to the equation:
$\theta =0.2\cos 8t,$ where $\theta$ is the angular displacement of the vertical in radians and $t$ is the time in seconds.
Calculate the maximum angular displacement and the reason of change of $\theta$ when $t=3$ seconds.

A) M.A.D $8t$ radians; R.O.C $0.2$rad/sec
B) M.A.D $0.2$ radians; R.O.C $8$ rad/sec
C) M.A.D $0.2$ radians; R.O.C $1.449$ rad/sec
D) none of above

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