Free Printable C1FE

1.

Complete the Pythagorean Theorem:

a)

(hypotenuse)^2 = (opposite side)^2 - (adjacent side)^2

b)

$(hypotenuse)^2 = (opposite side)^2 / (adjacent side)^2$

c)

(hypotenuse)^2 = (opposite side)^2 + (adjacent side)^2

d)

(hypotenuse)^2 = $(opposite side)^2 x (adjacent side)^2$

2.

Complete the Double Angle Formula: tan 2x = _______

a)

A. 2 tan x/(1 + tan2 x)

b)

B. tan2x/(1 - 2tanx)

c)

C. tanx/(1 - 2tan2x)

d)

D. $\frac{2\tan{x}}{1 - \tan^2{x}}$

3.

Complete the Power of Function: $cos^2 x$ = _______

a)

1/2 (1 + cos 2x)

b)

cos2x

c)

sin 2x

d)

1/2 (1 - cos 2x)

4.

Given the trigonometric function, tan y = x, what is the inverse of the function?

a)

$cot^{-1} x$

b)

$y = \cos^{-1} x$

c)

$tan^{-1} x$

d)

$y = \sin^{-1} x$

5.

Given the trigonometric function, sec y = x, what is the inverse of the function?

a)

$y = \tan^{-1} x$

b)

$sec^{-1} x$

c)

$csc^{-1} x$

d)

$y = \sin^{-1} x$

6.

What is the derivative of TanA?

a)

sinA/cosA

b)

$Sec^2A$

c)

$-\sin A / \cos^2 A$

d)

$Sec^2A$

7.

What is the derivative of CotA?

a)

$-\sin A / \cos^2 A$

b)

$-Csc^2A$

c)

sinA/cosA

8.

Which of the following a graph of y = sinA?

a)

A.

b)

B.

c)

C.

d)

D.

9.

Which of the following a graph of Tan A?

a)

A.

b)

B.

c)

C.

d)

D.

10.

Which of the following the appropriate formula for the first derivative of $y = x^2 \cot{x}$ ?

a)

Chain Rule

b)

Product Rule

c)

Quotient Rule

d)

Implicit Form

11.

Given a function in item number 24. Evaluate the first derivative.

a)

$y' = x(2cotA + xcsc^2A)$

b)

$y' = x(csc^2A + xcotA)$

c)

y' = $(2cotA + xcsc^2A)$

d)

$y' = -x(2csc^2A + xcotA)$

12.

Which of the following the 3rd derivative of y = sinax + cosbx?

a)

$y''' = -a^3Cosax + b^3Sinbx$

b)

$y''' = -a^3Cosax - b^3Sinbx$

c)

$y''' = a^3Cosax + b^3Sinbx$

d)

$y''' = a^3Cosax - b^3Sinbx$

13.

Which of the following the 1st derivative of a function $d/dy (Cosx/(3x-x^3))$ ?

a)

$(3x-x^3)(-sinx)+(cosx(3-3x^2))/(3x-x^3)^2$

b)

$(3x-x^3)(-sinx)+(cosx(3-3x^2))/(3x-x^3)^2$

c)

$(3x-x^3)(sinx)+(cosx(3-3x^2))/(3x-x^3)^2$

d)

$(3x+x^3)(-sinx)+(cosx(3+3x^2))/(3x-x^3)$

14.

The time rate defined as the quantity of x as a function of time. Which of the following symbol described the time rates.

a)

$d(x^2)/dt$

b)

dx/dt

c)

x/t

d)

dx

15.

A wall 4.0 meters high is 3.50 meters away from a building. Find the minimum length of a ladder that can reach the building with one end resting on the ground outside the wall. Show your solution.

a)

10.38 meters

b)

9.80 meters

c)

10.60 meters

d)

11.63 meters

16.

A balloon leaving the ground 20 meters from an observer has a rate of 1.5m/sec. How is the angle of elevation of the balloon increasing when it is 45 meters above the observer’s eyes?

a)

A. $\frac{d\theta}{dt} = 0.0124 \text{ rad/sec}$

b)

B. $\frac{d\theta}{dt} = 0.0134 \text{ rad/sec}$

c)

C. $\frac{d\theta}{dt} = 0.0142 \text{ rad/sec}$

d)

D. $\frac{d\theta}{dt} = 0.124 \text{ rad/sec}$

17.

Which of the following best described the value of the derivative of a Cotangent?

a)

A. $\csc^2 x$

b)

B. $-\csc^2 x$

c)

C. $\frac{\sin x}{\cos x}$

d)

D. $-\frac{\cos x}{\sin x}$

18.

Which of the following is the 1st derivative of $y = \sin ax + \cos ax$ ?

a)

A. $a^2 \cos ax - b^3 \sin bx$

b)

B. $a \cos ax - b \sin bx$

c)

C. $a^2 \cos ax - b^2 \sin bx$

d)

D. $a \cos ax + b \sin bx$

19.

Which of the following is the 3rd derivative of $y = \sin ax + \cos ax$ ?

a)

A. $-a^3 \cos ax + b^3 \sin bx$

b)

B. $a^3 \cos ax + b^3 \sin bx$

c)

C. $a^3 \cos ax - b^3 \sin bx$

d)

D. $-a^3 \cos ax - b^3 \sin bx$

20.

Which of the following is the appropriate method in finding the derivative of the function $\frac{d}{dy} \left( \frac{\cos x}{3x^2} \right)$ ?

a)

$\frac{vdu - udv}{v^2}$

b)

$\frac{vdu + udv}{v^2}$

c)

$udv + vdu$

d)

$udv - vdu$

21.

Which of the following best described the value of the derivative of a Cotangent?

a)

csc²x

b)

-csc²x

c)

cosx/sinx

d)

-cosx/sinx

22.

Which of the following best described for = aⁿ?

a)

Exponential Function

b)

Chain Rule

c)

Logarithmic Function

d)

Simple Function

23.

Which of the following an equivalent log₁₀x of common Logarithm?

a)

logx

b)

log10x

c)

10logx

d)

log(x/10)

24.

Which of the following the 1st derivative of y = eᵘ?

a)

dy/dx = eᵘ

b)

dy/dx = eᵘ

c)

du/dx = ue⁻ᵘ

d)

dy/dx = e⁴ᵘ

25.

Evaluate the 1st Derivative of y = ln(4x + 1)?

a)

y' = 4/(4x+1)

b)

y' = 1/(4x+1)

c)

y' = -4/(4x+1)

d)

y' = 4x/(4x+1)

26.

Evaluate the 2nd Derivative of y = eˣ + xlnx.

a)

y'' = eˣ + 1/x

b)

y'' = eˣ + ln(1/x)

c)

y'' = xeˣ + 1/x³

d)

y'' = eˣ + 3/x

27.

The population of a certain place is given by the equation P = $10,000e^{0.025t}$ , where t is the number of years after 1980. At what rate is the population expected to be growing in the year 2008?

a)

dp/dt = 503 people/year

b)

dp/dt = 508 people/year

c)

dp/dt = 506 people/year

28.

Which of the following is the general form of an exponential function?

a)

$y = a^n$

b)

y = ax + b

c)

y = log(a)

d)

y = n/a

29.

If $(a^{mn})*(a^n) = a^{(mn)}$ , $(x^2)^5$ , what is equal to?

a)

$x^7$

b)

$x^{25}$

c)

$x^{12}$

d)

$x^{10}$

30.

Which statement best describes the relationship between the natural exponential and the natural logarithm?

a)

The natural logarithm is the inverse function of the quadratic function.

b)

The natural exponential is the same as the logarithmic function.

c)

The natural exponential is the inverse function of the natural logarithm.

d)

They are unrelated functions.

31.

If $a^{-m} = 1/a^{m}$ , what is $2^{-3}$ equal to?

a)

1/6

b)

8

c)

-6

d)

1/8

32.

What is the derivative of the natural logarithmic function y = log u with respect to x?

a)

dx / du

b)

(du/dx) / u

c)

du / (u dx)

d)

u / (du/dx)

33.

Which rule is most appropriate to use when finding the derivative of y = ?

a)

Quotient rule

b)

Power rule

c)

Chain rule

d)

Product rule

34.

If y = $e^{(x+1)}$ , what is dy/dx?

a)

$x * e^{(x+1)}$

b)

$(x + 1)e^x$

c)

$e^{(x+1)}$

d)

$e^x$

35.

If y = log(cos(x)), what is the second derivative d²y/dx²?

a)

-sin(x)

b)

-sec2(x)

c)

-tan(x) * sec(x)

d)

-tan(x) * sec(x)

36.

Given y = $(x + 1)^x$ , which differentiation technique is most suitable for finding the second derivative?

a)

Product rule

b)

Power rule

c)

Quotient rule

d)

Logarithmic differentiation

37.

How do you locate the critical point and determine the maxima and minima of the curve?

a)

Set the first derivative equal to zero and analyze the second derivative

b)

Substitute x = 0

c)

Use the product rule

d)

Integrate the function

38.

To find the tangent and normal lines to the curve at point (1,1), which mathematical concept is primarily used?

a)

Integrals

b)

Sequences

c)

Derivatives

d)

Limits

39.

What is the inverse function of the natural logarithm?

a)

The logarithmic function y = log(x)

b)

The exponential function y = $a^n$

c)

The natural exponential function y = e^x

d)

The quadratic function y = x^2

40.

Explain the steps required to find the rate of change of population in the year 2008, given the equation $P = 10,000e^{0.025t}$ .

a)

Differentiate P with respect to t, substitute t = 8, and calculate the value.

b)

Integrate P with respect to t, substitute t = 8, and calculate the value.

c)

Substitute t = 8 directly into the equation without differentiation.

d)

Differentiate P with respect to t, substitute t = 0, and calculate the value.

41.

In a property of Logarithm, which of the following describes the equivalent of log_a(x/y)?

a)

A. log_a x - log_a y

b)

B. log_a x + log_a y

c)

C. log_a x / log_a y

d)

D. log_a x / log_a y

42.

Which of the following is the correct definition of the hyperbolic sine function, sinh(x)?

a)

$sinh(x) = (e^x - e^{-x})$

b)

sinh(x) = $(e^x + e^x)$

c)

$sinh(x) = \frac{(e^x - e^{-x})}{2}$

d)

$sinh(x) = \frac{(e^x + e^{-x})}{2}$

43.

The derivative of the hyperbolic sine function, y = sinh(u), with respect to x, is:

a)

cosh(u) * du/dx

b)

sinh(u) * du/dx

c)

cos(u) * du/dx

d)

sin(u) * du/dx

44.

Given the function y = cosh(u), what is its derivative with respect to x?

a)

-sinh(u) * du/dx

b)

tanh(u) * du/dx

c)

cosh(u) * du/dx

d)

sinh(u) * du/dx

45.

Given the identity sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y), which of the following is the correct expression for sinh(x + y)?

a)

sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y)

b)

sinh(x + y) = sinh(x)cosh(y) - cosh(x)sinh(y)

c)

sinh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y)

d)

sinh(x + y) = cosh(x)cosh(y) - sinh(x)sinh(y)

46.

Given sinh⁻¹(xy) = y cosh⁻¹ x, which differentiation technique is most appropriate to find y'?

a)

Direct substitution

b)

Implicit differentiation

c)

Partial differentiation

d)

Explicit differentiation

47.

Given z = t² sinh² t, which rule is most appropriate to find the second derivative d²z/dt²?

a)

Quotient rule only

b)

Product rule and chain rule

c)

No rule is needed

d)

Power rule only

48.

For tanh(x + y) = x² sinh y, which of the following is true about the differentiation process?

a)

A. Both x and y are constants

b)

B. Both x and y are treated as variables

c)

C. Only y is treated as a variable

d)

D. Only x is treated as a variable

49.

If y = cosh(4x + 1), what is dy/dx?

a)

4sinh(4x + 1)

b)

sinh(4x + 1)

c)

4cosh(4x + 1)

d)

cosh(4x + 1)

50.

Given the formula for the derivative of y = csch⁻¹(u), explain how you would use it to find the derivative of y = csch⁻¹(3x + 2) with respect to x.

a)

Substitute u = 3x + 2, find du/dx = 2, then use the formula -2 / ((3x + 2)√(1 + (3x + 2)²))

b)

Substitute u = 3x + 2, find du/dx = 3, then use the formula -3 / ((3x + 2)√(1 + (3x + 2)²))

c)

Substitute u = 3x + 2, find du/dx = 3, then use the formula 3 / ((3x + 2)√(1 + (3x + 2)²))

d)

Substitute u = 3x + 2, find du/dx = 1, then use the formula -1 / ((3x + 2)√(1 + (3x + 2)²))

51.

The derivative of y = tanh⁻¹(u) is du/dx divided by (1 - u²). Which relationship between tanh(y) and u explains this result?

a)

tanh(y) = u

b)

tanh(y) = 1/u

c)

tanh(y) = u²

d)

tanh(y) = 1 - u²

52.

Obtain the rectangular equations of the parametric equation x = sin t and y = cos t.

a)

$x^2 + y^2 = 1$

b)

$x^2 - y^2 = 1$

c)

$x^2 + y^2 = 0$

d)

$x^2 - y^2 = 0$

53.

Obtain the rectangular equation of the parametric equations x = t - 1 and y = 2t + 1.

a)

y = x - 3

b)

y = 2x − 1

c)

y = x + 2

d)

y = 2x + 3

54.

Find y' from x = 1 - t^2 and y = 3t + 2.

a)

-3 / 2t

b)

2t / 3

c)

-2t / 3

d)

3 / 2t

55.

Fill in the blank: The length of the arc PP' is ________.

a)

B. arc = radius × angle

b)

A. arc = diameter × angle

c)

C. arc = radius × diameter

d)

D. arc = angle ÷ radius

56.

Given the curve y = f(x), a circle and tangent line at P, the curvature of the curve at P is defined to be the curvature of that ________ or line.

a)

ellipse

b)

square

c)

circle

d)

parabola

57.

What is the ratio Δα/Δs used to represent in the context of curvature?

a)

The rate of change in direction of the curve per unit of arc

b)

The radius of curvature

c)

The tangent to the curve

d)

The length of the arc

58.

Find the equation of the circle.

a)

$x^2 - y^2 = r^2$

b)

$x^2 + y^2 + 2gx + 2fy + c = 0$

c)

$x^2 + y^2 = r^2$

d)

$x^2 + y^2 = 2r^2$

59.

Find the equation of the circle at (1,6)

a)

$(x - 1)^2 + (y - 6)^2 = r^2$

b)

$(x - 1)^2 + (y + 4)^2 = 25$

60.

Indeterminate forms are defined as, ____________________________.

a)

Forms that can be solved using basic arithmetic.

b)

Forms that are always undefined in mathematics.

c)

Expressions where the limit cannot be directly determined, such as 0/0 or ∞/∞.

d)

Expressions that always have a unique value.

61.

Determine the existence of indeterminate forms to functions.

a)

Indeterminate forms only exist in linear functions.

b)

Indeterminate forms always exist in all functions.

c)

Indeterminate forms never exist in functions.

d)

Indeterminate forms can exist in functions under certain conditions.

62.

Find the equation of the circle 3. y = x(x-3), at (-1,4)

a)

$(x + 1)^2 + (y - 4)^2 = 25$

b)

$(x - 1)^2 + (y - 4)^2 = 25$

c)

$(x - 1)^2 + (y + 4)^2 = 16$

d)

$(x + 1)^2 + (y - 4)^2 = 16$

63.

L’hopital’s Rule is defined as:

a)

A rule for integrating rational functions.

b)

A process for finding the area under a curve.

c)

A technique for solving quadratic equations.

d)

A method to evaluate limits of indeterminate forms using derivatives.

64.

The curvature and radius of a curve at a given point are related by which formula?

a)

Curvature = Radius

b)

Curvature = 1 / Radius

c)

Curvature = Radius^2

d)

Curvature = 2 × Radius

65.

Evaluate functions of indeterminate forms using the L’hopital’s Rule.

a)

By differentiating numerator and denominator and taking the limit.

b)

By substituting values directly.

c)

By integrating numerator and denominator.

d)

By multiplying numerator and denominator.

66.

Evaluate the following limit by applying the L'hopital's Rule: $lim_{x→0}(e^x + x)^{1/x}$

a)

1

b)

0

c)

e

d)

2

C1FE

Total questions: 66

Worksheet time: 50mins

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