Free Printable Math 1

1.

$\lim_{x\rightarrow11}\ \ \frac{x^2-121}{x-11}\ =\ ....$

a)

22

b)

21

c)

20

d)

19

e)

18

2.

$\lim_{x\rightarrow\infty}\ \frac{1-2x+2x^3}{x^3+x+1}=\ ...$

a)

- 4

b)

- 2

c)

1

d)

2

e)

$\infty$

3.

$\lim_{x\rightarrow\infty}\ \frac{x^4\ +3x^2+2}{x^3+x^2+1}\ =\ ...$

a)

0

b)

1

c)

2

d)

3

e)

$\infty$

4.

$\lim_{x\rightarrow\infty}\ \ \sqrt{4x^2-5x+7}-\sqrt{4x^2+7x-13}\ =\ ....$

a)

0

b)

$\infty$

c)

$\frac{1}{3}$

d)

3

e)

$-3$

5.

$\lim_{x\rightarrow-3}\ \ \frac{x+3}{x^2-9}\ =\ ....$

a)

0

b)

$-6$

c)

6

d)

$-\frac{1}{6}$

e)

$\frac{1}{6}$

6.

$\lim_{x\rightarrow-4}\ \ \ 17\ =\ ....$

a)

17

b)

$-68$

c)

$-4$

d)

$-17$

e)

0

7.

$\lim_{x\rightarrow-5}\ \ \ \frac{x+5}{2}\ =\ ....$

a)

0

b)

5

c)

$\infty$

d)

$\frac{5}{2}$

e)

$\frac{2}{5}$

8.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

9.

Let $\lim_{x\rightarrow-2}f\left(x\right)=16$ . Find $\lim_{x\rightarrow-2}\sqrt{f\left(x\right)}$

a)

4

b)

-2

c)

2

d)

16

10.

Find $\lim_{x\rightarrow0}\frac{\sqrt{1+x}-1}{\text{x}}$ .

a)

$\frac{1}{2}$

b)

$\frac{1}{4}$

c)

DNE

d)

0

11.

$\lim_{x\rightarrow0}3x+7$

a)

7

b)

10

c)

21

d)

None

12.

lim_x→0f(x)=

a)

1.5

b)

-3

c)

0

d)

Does not exist

13.

What is the limit of the function as x approaches -2?

a)

-2

b)

infinity

c)

-infinity

d)

DNE

14.

What is the limit as x approaches 3 from the left?

a)

negative infinity

b)

infinity

c)

3

d)

1

15.

Find the limit

(a)

16.

FInd the derivative:

f(x) = (-2/3)x³ - (1/2)x² + 9x

a)

-2x² - x + 9

b)

-2x² + x + 9

c)

2x² - x + 9

d)

2x² - x - 9

17.

Find the second derivative of the function:
f (x) = 2x - 5x⁶

a)

f ''(x)= 2 - 30x

b)

f ''(x) = 2-30x⁵

c)

f ''(x) = -30x⁵

d)

f ''(x) = -150x⁴

18.

Find the derivative.

a)

x⁴cosx - 4x³sinx

b)

x⁴cosx + 4x³sinx

c)

4x³cosx

d)

-4x³cosx

19.

Find the derivative of g(x)=(3x-2)/(x²+2)

a)

9x²+2

b)

3(x²+2)/(x²+2)²

c)

(-3x²+4x +6)/(x²+2)²

d)

(-3x²+10)/(x²+2)²

20.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

21.

Find y' if y=e^3x

a)

e^3x

b)

e^2x

c)

0

d)

3e^3x

22.

$derive\ y=\sin\left(5x\right)\ with\ respect\ to\ x$

a)

$-\cos\left(5x\right)$

b)

$\cos\left(5x\right)$

c)

$5\cos\left(5x\right)$

d)

$\sec\left(5x\right)$

23.

Find the derivative of
$f\left(x\right)=\ln\left(\cos\left(5x^2\right)\right)$

a)

$\frac{1}{\cos\left(5x^2\right)}$

b)

$-\frac{\left(10x\ \sin\left(5x^2\right)\right)}{\ \cos\left(5x^2\right)}$

c)

$-\frac{1}{10x\sin\left(5x^2\right)}$

d)

$\frac{-\sin\left(5x^2\right)}{\cos\left(5x_{ }^2\right)}$

24.

$y=x^2\sin x\ +\ \cos x$

Find y'.

a)

$y'=\left(2x-1\right)\sin x+x^2\cos x$

b)

$y'=x\sin x+\cos x$

c)

$y'=2x\cos x-\sin x$

d)

$y'=2x\sin x+\cos x-x^2\sin x$

25.

Find the derivative f(x) = (3x + 5)⁴

a)

f'(x) = 12(3x)³

b)

f'(x) = 12(3x + 5)³

c)

f'(x) = (3)⁴

d)

f'(x) = (3x + 5)³

26.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

27.

Find the derivative of f(x) = 6x³⁰-2x¹⁵ + 4x³ - 2x + 1

a)

f'(x) = 18x²⁹+ 30x¹⁵ + 12x³

b)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x²

c)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x² - 2

d)

f'(x) = 180x²⁹ - 30x¹⁴+ 12x² +1

28.

Find the derivative f(x) = xe^x

a)

f'(x) = e^x

b)

f'(x) = xe^x + xe^x

c)

f'(x) = e^x - xe^x

d)

f'(x) = e^x + xe^x

29.

Find the derivative:

y=log₉x

a)

y′=1 /((log9)x)

b)

y′=(ln9)/x

c)

y′=1/(x(lnx))

d)

y′=1/((ln9)x)

30.

a)

A

b)

B

c)

C

d)

D

31.

What are the local maxima of this graph?

a)

local maximum at (2.2, 3.9)

b)

local maximum at (-8, 5)

c)

local maximum at (5, -8)⋃(3.9, 2.2)

d)

local maximum at (-8, 5)⋃(2.2, 3.9)

32.

When is this function increasing?

a)

$\left(-2.5,2.5\right)$

b)

$\left(0,5\right)$

c)

$\left(0,55\right)$

d)

$\left(-5,0\right)$

33.

The blue dot on this graph represents a(n)...

a)

Absolute Maximum

b)

Absolute Minimum

c)

Relative Maximum

d)

Relative Minimum

34.

4 is a(n)...

a)

Absolute Maximum

b)

Absolute Minimum

c)

Local Maximum

d)

Local Minimum

35.

Find the local maximum and minimum

a)

Local max at x = -3

Local min at x = -3

b)

Local max at x = 1

Local min at x = -3

c)

Local max at x = -3

Local min at x = 1

d)

$\left(-\infty,\ \infty\right)$

36.

Given a function, f(x), if f'(x)>0 over a certain interval, then f(x) is __________ over that interval.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

37.

Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has _____________ at x.

a)

an inflection point

b)

a critical point

c)

a minimum

d)

a maximum

38.

What is the maximum value of f(x) = x³ - 3x² - 1 on the interval [-3, 2]?

a)

0

b)

-1

c)

2

d)

5

39.

What is a point of inflection?

a)

When a function goes from increasing to decreasing

b)

When a function goes from concave up to concave down

40.

If a function has a second derivative that is positive, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The function is concave up

d)

The function is concave down

41.

Let f(x) = $x^3 - 3x$ on the interval $[-2,2]$ . How many points c in $(-2,2)$ satisfy the conclusion of the Mean Value Theorem?

a)

0

b)

1

c)

2

d)

3

42.

Let f be continuous on $[a,b]$ and differentiable on $(a,b)$ . If $f'(x) > 0$ for all $x \in (a,b)$ , which statement is necessarily true?

a)

The MVT point c is unique

b)

f is linear

c)

$f(a) = f(b)$

d)

The MVT does not apply

43.

Suppose f satisfies the hypotheses of the Mean Value Theorem on $[a,b]$ and $f'(x) = 0$ for all $x \in (a,b)$ . What can be concluded?

a)

f has exactly one MVT point

b)

f is constant on $[a,b]$

c)

f is discontinuous

d)

The MVT fails

44.

Evaluate conceptually: $\lim_{x \to 0^+} x\ln x$ .

a)

0

b)

1

c)

$-\infty$

d)

The limit does not exist

45.

Which limit requires repeated application of L’Hôpital’s Rule to resolve?

a)

$\lim_{x\to 0} \frac{\sin x}{x}$

b)

$\lim_{x\to \infty} \frac{e^x}{x}$

c)

$\lim_{x\to 0} \frac{1 - \cos x}{x^2}$

d)

$\lim_{x\to \infty} \frac{\ln x}{x}$

46.

Consider $\lim_{x\to 0} \frac{e^x - 1 - x}{x^2}$ . Which statement is correct?

a)

L’Hôpital’s Rule cannot be applied

b)

One application of L’Hôpital’s Rule is sufficient

c)

Two applications of L’Hôpital’s Rule are required

d)

The limit diverges

47.

Suppose repeated applications of L’Hôpital’s Rule always yield an indeterminate form of $0/0$ . What is the most mathematically sound conclusion?

a)

The limit must be zero

b)

The limit does not exist

c)

L’Hôpital’s Rule is inconclusive and another method is needed

d)

The functions are identical

48.

Let f and g be differentiable and $\lim_{x\to a} \frac{f(x)}{g(x)} = L$ , but $\lim_{x\to a} f(x)g(x)$ does not exist. What does this imply?

a)

L’Hôpital’s Rule was applied correctly

b)

One of the hypotheses of L’Hôpital’s Rule failed

c)

The limit must equal $L$

d)

The Mean Value Theorem applies instead

49.

Which statement correctly compares the Mean Value Theorem and L’Hôpital’s Rule?

a)

Both require only continuity

b)

Both guarantee a numerical limit

c)

L’Hôpital’s Rule relies on ideas from the Mean Value Theorem

d)

The Mean Value Theorem is a special case of L’Hôpital’s Rule

50.

Which function pair best illustrates the connection between repeated L’Hôpital’s Rule and Taylor series?

a)

$\sin x$ and $\sin x$

b)

$e^x - 1$ and $x$

c)

$\ln x$ and $x$

d)

$x^2$ and $x^2$

Math 1

Total questions: 50

Worksheet time: 2hrs 51mins

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