CE Board Exam Randomizer

⬅ Back to Differential Calculus Topics

Linear Approximation and Differentials

$$dy = f'(x)\, dx$$
$$\Delta y \approx dy = f'(x)\Delta x$$
$$L(x) = f(a) + f'(a)(x-a)$$

Finding the Differential — CE Board

Find $dy$ if $y = x^3 - 4x + 1$.

The differential is $dy = f'(x)\,dx$.

$f'(x) = 3x^2 - 4$
$\boxed{dy = (3x^2 - 4)\,dx}$

Approximating a Square Root — CE Board

Use differentials to approximate the change in $y = \sqrt{x}$ as $x$ increases from 25 to 26.

$f(x) = \sqrt{x}$, $f'(x) = \dfrac{1}{2\sqrt{x}}$, $a = 25$, $dx = 1$.

$dy = f'(25)\cdot 1 = \dfrac{1}{2\sqrt{25}} = \dfrac{1}{10} = \boxed{0.1}$

So $\sqrt{26} \approx 5 + 0.1 = 5.1$. (Actual: $\sqrt{26} \approx 5.0990$)

Linear Approximation of a Cube Root — CE Board

Approximate $\sqrt[3]{8.1}$ using the linearization $L(x) = f(a) + f'(a)(x - a)$.

Let $f(x) = x^{1/3}$, $a = 8$.

$f(8) = 2,\quad f'(x) = \tfrac{1}{3}x^{-2/3},\quad f'(8) = \tfrac{1}{3\cdot 4} = \tfrac{1}{12}$
$L(8.1) = 2 + \dfrac{1}{12}(8.1 - 8) = 2 + \dfrac{0.1}{12} \approx \boxed{2.0083}$

(Actual: $\sqrt[3]{8.1} \approx 2.00829$)

Error Propagation in Surface Area — CE Board

The radius of a sphere is measured as $r = 6$ cm with a possible error of $dr = 0.05$ cm. Estimate the relative percentage error in the surface area $S = 4\pi r^2$.

$dS = 8\pi r\,dr$

Relative error:

$\dfrac{dS}{S} = \dfrac{8\pi r\,dr}{4\pi r^2} = \dfrac{2\,dr}{r} = \dfrac{2(0.05)}{6} = \dfrac{0.1}{6} \approx 0.0167$
$\text{Percentage error} \approx \boxed{1.67\%}$

Differential of a Product — CE Board

Find $dy$ if $y = x^2 \sin x$.

Apply the product rule:

$\dfrac{dy}{dx} = 2x\sin x + x^2 \cos x$
$\boxed{dy = (2x\sin x + x^2\cos x)\,dx}$
Scroll to zoom

Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: t804

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

The side of a square decreases at a rate of 4 cm/min. What is the change in area?

  1. -2P
  2. -4P
  3. -P
  4. -6P
For a square, $A = s^2$, so $\frac{dA}{dt} = 2s\frac{ds}{dt}$.
With $\frac{ds}{dt} = -4$:
$\frac{dA}{dt} = 2s(-4) = -8s$
Since the perimeter is $P = 4s$, this is $-8s = -2(4s)$:
$\boxed{\frac{dA}{dt} = -2P}$

Question Bank: t805

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

A stone dropped into a calm lake causes a series of circular ripples. The radius of the outer one increases at 3 m/sec. How rapidly is the disturbed area changing at the end of 3 seconds, in m^2/sec?

  1. 170
  2. 180
  3. 160
  4. 150
$\frac{dr}{dt} = 3$ m/s, so at $t = 3$ s the radius is $r = 9$ m.
$A = \pi r^2 \Rightarrow \frac{dA}{dt} = 2\pi r\frac{dr}{dt}$
$\frac{dA}{dt} = 2\pi(9)(3) = 54\pi$
$\boxed{170 \text{ m}^2/\text{s}}$

Question Bank: t810

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

The radius of an expanding sphere changes at the rate of 2 cm per minute. How fast is the surface area of the sphere changing when the radius is 25 cm, in cm^2/min.

  1. 1256.64
  2. 628.32
  3. 1856.32
  4. 963.21
Surface area $A = 4\pi r^2$, so $\frac{dA}{dt} = 8\pi r\frac{dr}{dt}$.
With $r = 25$ cm and $\frac{dr}{dt} = 2$ cm/min:
$\frac{dA}{dt} = 8\pi(25)(2) = 400\pi$
$\boxed{1256.64 \text{ cm}^2/\text{min}}$

Question Bank: t811

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

The volume of a cube is increasing at the rate of 21 cc/min. At what rate is its surface area increasing when the side is 80 cm long?

  1. 1.86 cm^2/min
  2. 2.14 cm^2/min
  3. 1.05 cm^2/min
  4. 1.32 cm^2/min
From $V = s^3$: $\frac{dV}{dt} = 3s^2\frac{ds}{dt}$.
$21 = 3(80)^2\frac{ds}{dt} \Rightarrow \frac{ds}{dt} = \frac{21}{19200}$ cm/min
Surface area $A = 6s^2$, so $\frac{dA}{dt} = 12s\frac{ds}{dt} = 12(80)\left(\frac{21}{19200}\right)$
$\boxed{1.05 \text{ cm}^2/\text{min}}$

Question Bank: t813

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

A rectangular parallelepiped with an altitude of 10 cm have its width decreasing by 2 cm/min and length increasing at 8 cm/min. How fast is the total surface area changing when its width is 16 cm and length is 22 cm?

  1. +224 cm^2/min
  2. -312 cm^2/min
  3. -242 cm^2/min
  4. +288 cm^2/min
With constant height $h = 10$, total surface area $S = 2(Lw + Lh + wh)$.
$\frac{dS}{dt} = 2\left(L\frac{dw}{dt} + w\frac{dL}{dt} + h\frac{dL}{dt} + h\frac{dw}{dt}\right)$
At $w = 16$, $L = 22$, $\frac{dw}{dt} = -2$, $\frac{dL}{dt} = 8$:
$\frac{dS}{dt} = 2\left(22(-2) + 16(8) + 10(8) + 10(-2)\right) = 2(144)$
$\boxed{+288 \text{ cm}^2/\text{min}}$

Question Bank: t835

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

The stiffness of a rectangular timber is proportional to the width and the cube of the depth. The depth of the stiffest beam that can be made of a circular log whose diameter is 20 inches is:

  1. 17.32 inches
  2. 10 inches
  3. 20 inches
  4. 15.87 inches
Stiffness $S = k\,w\,d^3$ with the beam inscribed in the log: $w^2 + d^2 = 20^2 = 400$.
$S = k\,w(400 - w^2)^{3/2}$
$\frac{dS}{dw} = k(400 - w^2)^{1/2}(400 - 4w^2) = 0 \Rightarrow w^2 = 100$
$d^2 = 400 - 100 = 300 \Rightarrow d = \sqrt{300}$
$\boxed{d = 17.32 \text{ inches}}$

Question Bank: t905

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

A hemispherical tank of radius 1.5 meters is filled with 3000 liters of water.

Find the depth of water in the tank in meters.

  1. 0.654
  2. 0.105
  3. 1.234
  4. 0.891

Find the surface area of the tank in contact with water in square meter.

  1. 8.398
  2. 10.234
  3. 7.334
  4. 5.872

Find the work done in pumping all the water to the top of the tank in KJ.

  1. 27.21
  2. 33.05
  3. 22.12
  4. 35.86

Part 1.

Water volume = 3000 L = 3 m3. At depth $h$ in a hemispherical bowl ($R = 1.5$), the volume is $V = \pi\left(Rh^2 - \frac{h^3}{3}\right)$:
$\pi\left(1.5h^2 - \frac{h^3}{3}\right) = 3 \Rightarrow 1.5h^2 - \frac{h^3}{3} = 0.955$
Solving:
$\boxed{h = 0.891 \text{ m}}$

Part 2.

The wetted surface is the spherical zone of height $h$:
$A = 2\pi R h = 2\pi(1.5)(0.891)$
$\boxed{8.398 \text{ m}^2}$

Part 3.

A slice at height $z$ above the bottom has volume $\pi(2Rz - z^2)\,dz$ and is lifted $(R - z)$ to the rim.
$W = \rho g\pi\int_0^{0.891}(3z - z^2)(1.5 - z)\,dz$
$= 9810\pi(0.883)$
$\boxed{27.21 \text{ kJ}}$

Question Bank: t910

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles every two days. If there are 20,000 mosquitoes in the area initially, how many mosquitoes are there after 8 days?

  1. 240,000
  2. 320,000
  3. 160,000
  4. 480,000
Exponential growth doubling every 2 days. In 8 days there are $\frac{8}{2} = 4$ doublings:
$P = 20{,}000\times 2^4 = 20{,}000\times 16$
$\boxed{320{,}000}$

Question Bank: t1400

MSTE - Differential Calculus / Differentials / BEMz

An elliptical plot of garden has a semi-major axis of 6m and a semi-minor axis of 4.8meters. If these are increased by 0.15m each, find by differential equations the increase in area of the garden in sq.m.

  1. $0.62 \pi$
  2. $1.62\pi$
  3. $2.62\pi$
  4. $2.62\pi$

Solution pending in psadquestions/t1400.json.

Question Bank: t1401

MSTE - Differential Calculus / Differentials / BEMz

The diameter of a circle is to be measured and its area computed. If the diameter can be measured with a maximum error of 0.001cm and the area must be accurate to within 0.10sq.cm. Find the largest diameter for which the process can be used.

  1. 64
  2. 16
  3. 32
  4. 48

Solution pending in psadquestions/t1401.json.

Question Bank: t1402

MSTE - Differential Calculus / Differentials / BEMz

The altitude of a right circular cylinder is twice the radius of the base. The altitude is measured as 12cm. With a possible error of 0.005cm, find the approximately error in the calculated volume of the cylinder.

  1. 0.188 cu cm
  2. 0.144 cu cm
  3. 0.104 cu cm
  4. 0.126 cu cm

Solution pending in psadquestions/t1402.json.

Question Bank: t1403

MSTE - Differential Calculus / Differentials / BEMz

What is the allowable error in measuring the edge of a cube that is intended to hold a cu m, if the error in the computed volume is not to exceed 0.03 cu m?

  1. 0.002
  2. 0.0025
  3. 0.003
  4. 0.001

Solution pending in psadquestions/t1403.json.

Question Bank: t1404

MSTE - Differential Calculus / Differentials / BEMz

If $y=x^{3/2}$ what is the approximate change in y when x changes from 9 to 9.01?

  1. 0.045
  2. 0.068
  3. 0.070
  4. 0.023

Solution pending in psadquestions/t1404.json.

Question Bank: t1405

MSTE - Differential Calculus / Differentials / BEMz

The expression for the horsepower of an engine is $P=0.4$ n $x^{2}$ where n is the number of cylinders and x is the bore of cylinders. Determine the power differential added when four cylinder car has the cylinders rebored from 3.25cm to 3.265cm.

  1. 0.156 hp
  2. 0.210 hp
  3. 0.319 hp
  4. 0.180 hp

Solution pending in psadquestions/t1405.json.

Question Bank: t1406

MSTE - Differential Calculus / Differentials / BEMz

A surveying instrument is placed at a point 180m from the base of a bldg on a level ground. The angle of elevation of the top of a bldg is 30 degrees as measured by the instrument. What would be error in the height of the bldg due to an error of 15minutes in this measured angle by differential equation?

  1. 1.05m
  2. 1.09m
  3. 2.08m
  4. 1.05m

Solution pending in psadquestions/t1406.json.

Question Bank: t1407

MSTE - Differential Calculus / Differentials / BEMz

If $y=3x^{2}-x+1$, find the point x at which dy/dx assume its mean value in the interval $x=2$ and $x=4$.

  1. 3
  2. 6
  3. 4
  4. 8

Solution pending in psadquestions/t1407.json.

Question Bank: t1408

MSTE - Differential Calculus / Differentials / BEMz

Find the approximate increase by the use of differentials, in the volume of the sphere if the radius increases from 2 to 2.05.

  1. 2.51
  2. 2.25
  3. 2.12
  4. 2.86

Solution pending in psadquestions/t1408.json.

Question Bank: t1409

MSTE - Differential Calculus / Differentials / BEMz

If the area of a circle is 64π sq mm, compute the allowable error in the area of a circle if the allowable error in the radius is 0.02 mm.

  1. 1.01 sq mm
  2. 1.58 sq mm
  3. 2.32 sq mm
  4. 0.75 sq mm

Solution pending in psadquestions/t1409.json.

Question Bank: t1410

MSTE - Differential Calculus / Differentials / BEMz

If the volume of a sphere is 1000π/6 cu mm and the allowable error in the diameter of the sphere is 0.03 mm, compute the allowable error in the volume of a sphere.

  1. 6.72 cu mm
  2. 4.71 cu mm
  3. 5.53 cu mm
  4. 3.68 cu mm

Solution pending in psadquestions/t1410.json.

Question Bank: t1411

MSTE - Differential Calculus / Differentials / BEMz

A cube has a volume of 1728 cu mm. If the allowable error in the edge of a cube is 0.04 mm, compute the allowable error in the volume of the cube.

  1. 17.28 cu mm
  2. 16.88 cu mm
  3. 15.22 cu mm
  4. 20.59 cu mm

Solution pending in psadquestions/t1411.json.