2.1 Differentiation and the Tangent Line

Is the above function differentiable for all values of x? Why?

• Yes, the slope of the tangent line can be calculated for all values of x.

• Yes, the function is continuous for all values of x so the function is differentiable.

• No, the function has a sharp turn in the first quadrant so if is not differentiable.

• No, the function is discontinuous so it is not differentiable.

Tags

CCSS.HSA.REI.D.10

Is the above function differentiable at

x = 0? Why?

• Yes. The derivative would be equal to zero since there is a vertical tangent.

• No. The derivative would not exist at x = 0 since there is a vertical tangent.

• No. The derivative does not exist since $\lim_{x\rightarrow0}\ x^{\frac{1}{3}}$ does not exist.

• Yes since $\lim_{h\rightarrow0}\ \frac{\left(x+h\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{h}$ exists.

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

For what value(s) of x is the function continuous but not differentiable?

• x = -1

• x = 0

• x = 2

• x = 3; x = -2

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

Find all points of the domain of  $y\ =\ \left|x-2\right|+3$  where  $\frac{dy}{dx}$  does not exist. 

•  $\frac{dy}{dx}$  exists for all real numbers.

•  $\frac{dy}{dx}$  does not exist at x = 0.

•  $\frac{dy}{dx}$  does not exist at x = 2.

•   $\frac{dy}{dx}$  does not exist at x = 3. 

Derivative means the same thing as

• slope of the tangent line
• slope of the normal line
• exponent
• potato

Tags

CCSS.HSS.ID.C.7

The derivative of a function is its

• Slope
• Maximum/Minimum
• Instantaneous rate of change
• Common Denominator

Tags

CCSS.HSF.IF.B.4

What is the equation of the line tangent to f(x)= 4x²+2x-1 at x=0?

• y+1=2(x+1)

• y=2x-1

• y=2x+2

• y= -x-1

Tags

CCSS.HSA.CED.A.1

CCSS.HSA.REI.B.3

If f '(x) = 0, what does that mean with respect to the tangent line?

• Tangent line is vertical

• Tangent line has a positive slope

• Tangent line is horizontal

• Tangent line has a negative slope

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

• The function is increasing
• The function is decreasing
• The concavity of the function is up
• The concavity of the function is down

Tags

CCSS.HSF.IF.B.4

Find the slope at x = 3 of the equation f(x) = 3x²

• 6x

• 27

• 18

• 3x

Tags

CCSS.8.EE.B.5