Mercer County Community College
Prof. Porter
MAT151
Exam 1
NAME:____________________
1. What is Calculus?
What is varying in calculus?
How are the variations described?
2. Use the δ-ε definition of limits to show that
lim4 x  3
x2
(find δ if ε = .01 so that the difference between the
limit and the function is less than ε when the x values are within δ of 2)
Answer:____________________
3. Evaluate the limit:
lim
x 3
x 2  5x  6
x2  9
Answer:____________________
Evaluate the more complicated limit:
 100 
cos

 x 
lim
x
x  
Answer:____________________
4. Explain on when the function is continuous . f ( x) 
x 2  5x  6
x2  9
Answer:____________________
Where is the function removably discontinuous?
Answer:____________________
Graph the function:
 Ax  7 x  2
5. Given the function, find A so that the function is continuous: f ( x)   2
x2
 x
Answer:____________________
Give a qualitative graph of the continuous function.
6. P(t) = t 2(5 – t) + 3 represents the height of a ball at time t.
Evaluate the function at 0 and 1 and use that information to find the average speed of the ball
between t = 0 and t = 2.
Answer:_________________
Find the instantaneous speed at t = 1 for this problem.
Answer:_________________
7. Use the definition of derivative to find f’(x) if f(x) = x( x + 1)
Be sure to give f(x+h) clearly.
Answer:____________________________
(No credit for just answer- must use definition!)
8. Find f’ of the following:
f ( x)  x  sin(2 )
Answer:____________________________
f ( x) 
cos( x)
x3
Answer:____________________________
9. Find g’(x) g ( x)  x3e2 x
Answer:_____________________________
Find g’’(x)
Answer:____________________________
10. What is another way of saying the slope of the tangent line?
Find f’(0) for the equation f(x) = x(x+1)
Find the equation of the line tangent to the curve y = x(x+1) when x = 0 and y = 1
Answer:____________________________
11. Find y’ if y =
sin(4 x  7)
Which rules did you use (be sure to name them)?
x
Rules:_____________________
Answer:____________________
Find y’ if y = (ln x)(sin 1 (e2 x )) Which rules did you use (be sure to name them)?
Rules:_____________________
Answer:____________________
12. Evaluate the limits.
.2 x 20  5 x  6
a. lim
x 
x 20  9
Answer:____________________
.2 x 200  5 x  6
x 
x 20  9
b. lim
Answer:____________________
.2 x  5 x  6
x 
x 200  9
20
c. lim
Answer:____________________
.2 x  5 x  6
x 20  9
21
d. lim
x 
Answer:____________________