MATH 250 - Calculus 1: Practice Test 1 PDF

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These are practice tests of the 4 exams in Calculus 1 at SBVC....

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Calculus Practice Test #1

1. Consider the function

f  x  13x  x 2

and the point

P  6, 42 

slope of the secant line passing through P  6, 42  and answer to one decimal place.

on the graph of f. Find the

Q x, f  x 

for x 2 . Round your

2. Use the rectangles in the graph given below to approximate the area of the region bounded by y 5 x , y 0, x 1, and x 5. Round your answer to three decimal places.

3.

Determine the following limit. (Hint: Use the graph to calculate the limit.) lim  6  x  x 1

4.

Find the limit. lim x 5

x + 139 x2

5.

Let  x 2  5, x  1 f ( x)  x 1  1, .

Determine the following limit. (Hint: Use the graph to calculate the limit.) lim f ( x) x 1

6.

Find the limit L. Then use the ε-δ definition to prove that the limit is L. lim ( 2 x +1) x →6

7.

8.

2 lim g( f ( x)) Let f ( x) 7 + 4 x and g ( x)  x + 3 . Find x 5

 5x  lim tan   x   3  Find

9.

Find the limit (if it exists). x –1  2 x 5

lim x 5

lim

10. Find

 x 0

f ( x  x)  f ( x) x where f(x) = x2 – 3x.

11. Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = 4.

(i)

lim f ( x)

x  4

(ii)

lim f ( x)

x  4

(iii)

lim f (x ) x4

12. Use the graph to determine the following limits, and discuss the continuity of the function at x = -4.

(i)

lim f ( x )

x  –4

(ii)

lim f ( x )

x  –4

(iii)

lim f (x )

x  –4

13. Find the limit (if it exists). lim

x  49

x 7 x  49

f (x ) 

14. Find the x-values (if any) at which the function of the discontinuities are removable?

15.

Find the constant a such that the function

sin x  , x0  –9  f (x )  x  a – 3x , x  0 

is continuous on the entire real line.

x + 10 x + 7x – 30 is not continuous. Which 2

16. Find the value of c guaranteed by the Intermediate Value Theorem. f (x ) x2 – x – 9,  4, 8  , f (c) 11

.

f (x ) 

17. Find all the vertical asymptotes (if any) of the graph of the function

lim

18. Find

x  8

x– 6 –x + 8

4 x x  3  x 2

.

19. A 35-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of r

2x 1225  x 2

ft/sec

where x is the distance between the base of the ladder and the house. Find the rate r when x is 28 feet.

20.

lim x →0

Find the limit if it exists:

√2+x−√ 2 x

sin 4 x Find x →0 7 x lim

21.

CALCULUS TEST 1 SOLUTIONS 1. Consider the function

f  x 13x  x 2

secant line passing through place.

P 6, 42

P 6, 42 and the point  on the graph of f. Find the slope of the

and

Q  x , f  x 

m= We can use the familiar slope formula:

for x 2 . Round your answer to one decimal

y 2− y 1 x 2−x 1

m=

42−22 20 = =5 . 0 4 6−2

2. Use the rectangles in the graph given below to approximate the area of the region bounded by y 5 x , y 0, x 1, and x 5. Round your answer to three decimal places.

There are four rectangles; each has a width of 1, which is nice for us! The heights are simply f(1), f(2), f(3), and f(4). So the total area is:

(

1

3.

)

5 5 5 5 + + + ≈10 . 417 1 2 3 4

Determine the following limit. (Hint: Use the graph to calculate the limit.) lim  6  x  x 1

Just by looking at the graph, it approaches 5 from both sides. So 5 is the limit.

4.

Find the limit.

By plugging in 5, the limit is 12/3, which equals 4.

x + 139 lim x 5 x2

5.

Let  x 2  5, x  1 f ( x)  x 1  1, .

Determine the following limit. (Hint: Use the graph to calculate the limit.) lim f ( x) x 1

Even though f(1) is 1, not six, the graph approaches 6 from both sides. Hence the limit is 6.

6.

Find the limit L. Then use the ε-δ definition to prove that the limit is L. lim ( 2 x +1) x →6

Limit is 13 (by plugging in). We decide to let δ = ε/2 (see from left row).

|F(x )−L|...