Free Printable FINAL EXAM IN BASIC CALCULUS

1.

Who was the first person to formulate the Chain Rule?

a)

Isaac Newton

b)

Leonhard Euler

c)

Gottfried Leibniz

d)

James Gregory

2.

Use the Chain Rule to find the derivative of the function $y=\left(\frac{x}{x-1}\right)^6$

a)

$y'=-\ \frac{6x^5}{\left(x-1\right)^7}$

b)

$y'=\ \frac{6x^5}{\left(x-1\right)^7}$

c)

$y'=-\ \frac{x^5}{\left(x-1\right)^7}$

d)

$y'=\frac{x^5}{\left(x-1\right)^7}$

3.

Find the derivative of $y=2x^4+\sec x+8\cos x$

a)

$y'=8x^3-\sec x\tan x+8\sin x$

b)

$y'=8x^3+\sec x\tan x+8\sin x$

c)

$y'=8x^3+\sec x\tan x-8\sin x$

d)

$y'=-8x^3-\sec x\tan x+8\sin x$

4.

A car is travelling in a rectilinear movement with an equation of $𝑦=2x^3−\ \frac{5}{2}x^2−6𝑥+1$ . At what time is the velocity zero?

a)

$t=\pm\frac{2}{3}$

b)

$t=\frac{3}{2},\ t=-\frac{2}{3}$

c)

$t=\pm\frac{3}{2}$

d)

$t=-\frac{3}{2},\ t=\frac{2}{3}$

5.

Find the limit at infinity

a)

0

b)

1

c)

infinity

d)

DNE

6.

A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection?

a)

57.60

b)

57.88

c)

59.20

d)

67.40

7.

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

v = 0

b)

v = 1

c)

v = -1

d)

v = -2

8.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12

9.

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the height of the water changing when the water is 2 ft deep?

a)

-9/(8pi) ft/sec

b)

9/(8pi) ft/sec

c)

-8/(9pi) ft/sec

d)

8/(9pi) f/tsec

10.

$\int\left(5\sqrt{x^3}-16e^{-4x}+\frac{1}{x}\right)dx$

a)

$2x^{\frac{5}{2}}+4x^{-4x}+\ln\left|x\right|+C$

b)

$5x^{\frac{1}{3}}-4e^{-4x}+1+C$

c)

$\frac{5}{2}x^{\frac{3}{2}}-16e^{-4x}+\ln\left|x\right|+C$

d)

Got lazy with fake answers

11.

$\int4\sin\left(-x\right)dx$

a)

$4\cos\left(x\right)+C$

b)

$-4\sin\left(-x\right)+C$

c)

$4\cos\left(-x\right)+C$

d)

$-4\cos\left(-x\right)+C$

12.

$\int\left(5x^2-7x+6\right)dx$

a)

$\frac{5}{3}x-\frac{7}{2}x+6x+C$

b)

$x+x+x+x+x+C$

c)

$x^3-3x^2+6x+C$

d)

$\frac{5}{3}x^3-\frac{7}{2}x^2+6x+C$

13.

$\int\left(6\sqrt{x}-1\right)dx$

a)

$6x^{\frac{3}{2}}+x+C$

b)

$4x^{\frac{3}{2}}-x+C$

c)

$3x^{-\frac{1}{2}}-x+C$

d)

I didn't look at my notes to see how to do this one.

14.

Integrate $\sqrt{x}$ with respect to x

a)

$x^{\frac{1}{2}}+c$

b)

$\frac{1}{2}x^{-\frac{1}{2}}+\ c$

c)

$\frac{2}{3}x^{\frac{3}{2}}+\ c$

d)

$\frac{3}{2}x^{\frac{3}{2}}+\ c$

15.

If y = $ax^n$ , then $\int_{ }^{ }y\ dx\ =$

a)

$\frac{ax^{n+1}}{n+1}$

b)

$\frac{ax^{n-1}}{n-1}+\ c$

c)

$\frac{ax^{n+1}}{n}+\ c$

d)

$\frac{ax^{n+1}}{n+1}+\ c$

16.

∫(cos x + 3x²)dx =

a)

-sin x + x³ + c

b)

sinx + x³ + c

c)

-sin x + 6x

d)

🌞

17.

Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by dy/dt = ky. There was initially 10,000 ft^3 at t=0. After 4 hours there were 8000 ft^3 remaining.
What is the value of k in the differential equation (calculator)

a)

-0.050

b)

-0.056

c)

-.169

d)

-.200

18.

Find the mistake if possible:

a)

The Diff EQ is solved correctly

b)

Step 1 is incorrect. The separation of variables wasn't done correctly.

c)

Step 2 is incorrect. They didn't integrate x² correctly.

d)

Step 3 is incorrect. They didn't take the reciprocal of x³/3+C correctly.

19.

Find the Particular Solution

a)

$y=2+e^{\left(\frac{x^2}{2}+x\right)}$

b)

$y=2e^{\left(\frac{x^2}{2}+x\right)}$

c)

$y=\ln\left|\frac{x^2}{2}+x+1\right|+2$

d)

$y=\ln\left|\frac{x^2}{2}+x+e^2\right|$

20.

A lump of radioactive material has a mass of X(t). Its mass slowly decays over time. What is its differential equation?

a)

W

b)

X

c)

Y

d)

Z

21.

Water is pumped from a lake at a constant rate. The volume of water in the lake is X(t). What is the differential equation for this?

a)

W

b)

X

c)

Y

d)

Z

22.

The degree of a differential equation is define by a

a)

Positive real number

b)

Positive rational number

c)

Positive integer

d)

All the above

23.

Solution of a linear differential equation of first order can be obtain by

a)

multiplying on both the sides by its integrating factor.

b)

adding on both the sides by integrating factor.

c)

subtracting integrating factor from both sides

d)

None of the above

24.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

$y=\ln\left|15t\right|$

b)

$y=16t^3$

c)

$y\ =\ \sqrt{2t^3-16}$

d)

$y=2t^3-16$

25.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{x}{y}$

a)

$\ln y=\frac{x^2}{2}+C$

b)

$y=\frac{x^2}{2}+C$

c)

$x^2=y^2+C$

d)

$y^2=x^2+2C$

26.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$

a)

$\sin y=1+C$

b)

$\sin y=x+C$

c)

$-\sin y=\frac{x^2}{2}+C$

d)

$\sin y=0+C$

27.

Dy/dx= tanx+15x2+e^x+1/x

a)

y= -ln|sinx| + 5x³ + e^x + ln|x| + c

b)

y= secxtanx + 5x³ + e^x + ln|x| + c

c)

y= -ln|cosx| + 5x³ + e^x + ln|x| + c

d)

y= sec²(x) + 5x³ + e^x + ln|x| + c

28.

What is the approximate area under the curve, using 4 intervals with heights using left values?

a)

20

b)

14

c)

10

d)

8

29.

a)

2955

b)

7260

c)

7150

d)

6900

e)

7520

30.

What does picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

31.

For a function that is strictly decreasing, a right hand Riemann Sum is which of the following:

a)

Overestimate

b)

Underestimate

c)

Exact Solution

d)

Unable to Determine

32.

For a function that is strictly decreasing, a trapezoidal approximation will be which of the following:

a)

Underestimate

b)

Overestimate

c)

Exact Solution

d)

Unable to Determine

33.

Use a midpoint Riemann Sum to approximate the area between 0 to 3 with 3 subintervals.

a)

14

b)

7

c)

26

d)

11

34.

Find the Left-hand Riemann Sum, with three sub-intervals indicated by the table.

a)

28

b)

16

c)

34

d)

23

35.

The graph of a piecewise linear function f for $-1\le x\le4$ is shown. What is the value of $\int_{-1}^4f\left(x\right)dx?$

a)

1

b)

4

c)

8

d)

2.5

e)

5.5

36.

$\int_{\frac{1}{2}}^5\left(2-\frac{1}{\left(x\right)}\right)dx$

a)

9+ln10

b)

0

c)

5+ln(1/2)

d)

9-ln10

37.

$\ Evaluate\ \ \ \int_0^{10}\left(\frac{1}{\left(3+2\sin x\right)}\right)dx$ using your calculator

a)

12.324

b)

7.602

c)

3.802

d)

17.987

38.

$\int_3^3\left(x^2+2x+1\right)\ dx$ =

a)

3

b)

0

c)

6

d)

-1

39.

a)

12/3

b)

6.5

c)

5.5

d)

11/3

40.

$\int_1^5\left(-x^2+6x-10\right)dx$

a)

$\frac{28}{3}$

b)

$\frac{-28}{3}$

c)

28

d)

$-\frac{7}{2}$

41.

a)

-2

b)

-1

c)

1

d)

2

42.

$\int_1^4\ \left|x\ -\ 3\right|\ dx$ =

a)

$\frac{-5}{2}$

b)

6

c)

$\frac{-3}{2}$

d)

$\frac{5}{2}$

43.

a)

1/3

b)

8/3

c)

16/3

d)

24/3

44.

a)

0

b)

1/2

c)

-1/2

d)

1

45.

Which of these are definite integrals?

a)

b)

c)

d)

46.

What should the value of "u" be equal to in this integral.

$\int\ \frac{\sin\left(\ln x\right)}{x}dx$

a)

$u=x$

b)

$u=\frac{\ln x}{x}$

c)

$u=\ln x$

d)

$u=\sin\left(\ln x\right)$

47.

What should "du" equal in this integral? $\int\ \frac{\tan x\sec x}{\sec^{5\ }x}dx$

a)

$du=\tan x\sec x\ dx$

b)

$du=\sec x\ dx$

c)

$du=\sec2x\ dx$

d)

$du=\tan x\ dx$

48.

Evaluate the Integral.

$\int\ \frac{e^{\sqrt{x}}}{\sqrt{x}}dx$

a)

$2e^{\sqrt{x}}+C$

b)

$\frac{1}{2}e^{\sqrt{x}}+C$

c)

$2\sqrt{x}e^{\sqrt{x}}+C$

d)

$e^{\sqrt{x}}+C$

FINAL EXAM IN BASIC CALCULUS

Total questions: 48

Worksheet time: 2hrs 57mins

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