Practice Test
For the following exercises, use the graph of f f in Figure 1.
f(1) f(1)
lim x→ −1 + f(x) lim x→ −1 + f(x)
lim x→ −1 − f(x) lim x→ −1 − f(x)
lim x→−1 f(x) lim x→−1 f(x)
lim x→−2 f(x) lim x→−2 f(x)
At what values of x x is f f discontinuous? What property of continuity is violated?
For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x x approaches a. a. If the function has a limit as x x approaches a, a, state it. If not, discuss why there is no limit
f(x)={ 1 x −3, if x≤2 x 3 +1,if x>2 a=2 f(x)={ 1 x −3, if x≤2 x 3 +1,if x>2 a=2
f(x)={ x 3 +1, if x<1 3 x 2 −1, if x=1 − x+3 +4, if x>1 a=1 f(x)={ x 3 +1, if x<1 3 x 2 −1, if x=1 − x+3 +4, if x>1 a=1
For the following exercises, evaluate each limit using algebraic techniques.
lim x→−5 ( 1 5 + 1 x 10+2x ) lim x→−5 ( 1 5 + 1 x 10+2x )
lim h→0 ( h 2 +25 −5 h 2 ) lim h→0 ( h 2 +25 −5 h 2 )
lim h→0 ( 1 h − 1 h 2 +h ) lim h→0 ( 1 h − 1 h 2 +h )
For the following exercises, determine whether or not the given function f f is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.
f(x)= x 2 −4 f(x)= x 2 −4
f(x)= x 3 −4 x 2 −9x+36 x 3 −3 x 2 +2x−6 f(x)= x 3 −4 x 2 −9x+36 x 3 −3 x 2 +2x−6
For the following exercises, use the definition of a derivative to find the derivative of the given function at x=a. x=a.
f(x)= 3 5+2x f(x)= 3 5+2x
f(x)= 3 x f(x)= 3 x
f(x)=2 x 2 +9x f(x)=2 x 2 +9x
For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.
f(x)=| x−2 |−| x+2 | f(x)=| x−2 |−| x+2 |
f(x)= 2 1+ e 2 x f(x)= 2 1+ e 2 x
For the following exercises, explain the notation in words when the height of a projectile in feet, s, s, is a function of time t t in seconds after launch and is given by the function s(t). s(t).
s(0) s(0)
s(2) s(2)
s'(2) s'(2)
s(2)−s(1) 2−1 s(2)−s(1) 2−1
s(t)=0 s(t)=0
For the following exercises, use technology to evaluate the limit.
lim x→0 sin(x) 3x lim x→0 sin(x) 3x
lim x→0 tan 2 (x) 2x lim x→0 tan 2 (x) 2x
lim x→0 sin(x)(1−cos(x)) 2 x 2 lim x→0 sin(x)(1−cos(x)) 2 x 2
Evaluate the limit by hand.
lim x→1 f(x), where f(x)={ 4x−7 x≠1 x 2 −4 x=1 lim x→1 f(x), where f(x)={ 4x−7 x≠1 x 2 −4 x=1
At what value(s) of x x is the function below discontinuous?
f(x)={ 4x−7x≠1 x 2 −4x=1 f(x)={ 4x−7x≠1 x 2 −4x=1
For the following exercises, consider the function whose graph appears in Figure 3.
Find the average rate of change of the function from x=1 to x=3. x=1 to x=3.
Find all values of x x at which f'(x)=0. f'(x)=0.
Find all values of x x at which f'(x) f'(x) does not exist.
Find an equation of the tangent line to the graph of f f the indicated point: f(x)=3 x 2 −2x−6, x=−2 f(x)=3 x 2 −2x−6, x=−2
For the following exercises, use the function f(x)=x ( 1−x ) 2 5 f(x)=x ( 1−x ) 2 5.
Graph the function f(x)=x ( 1−x ) 2 5 f(x)=x ( 1−x ) 2 5 by entering f(x)=x ( ( 1−x ) 2 ) 1 5 f(x)=x ( ( 1−x ) 2 ) 1 5 and then by entering f(x)=x ( ( 1−x ) 1 5 ) 2 f(x)=x ( ( 1−x ) 1 5 ) 2.
Explore the behavior of the graph of f(x) f(x) around x=1 x=1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x=1. x=1.
1. ਪਰਿਭਾਸ਼ਾ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਹੇਠ ਲਿਖੇ ਅਭਿਆਸਾਂ ਲਈ, ਹਰੇਕ ਫੰਕਸ਼ਨ ਦਾ ਡੈਰੀਵੇਟਿਵ ਲੱਭੋ: lim h→0 f(x+h)−f(x) h
2. f(x)=2x−8
3. f(x)=4 x 2 −7
4. f(x)=x− 1 2 x 2
5. f(x)= 1 x+2
6. f(x)= 3 x−1
7. f(x)=− x 3 +1
8. f(x)= x 2 + x 3
9. f(x)= x−1