AP Calculus District CA Review

​AP Calculus AB
District Assessment Review

​The graph of f(x) is provided

​find the value of

Multiple Choice

$$h\left(x\right)=\frac{5}{x^3}-f\left(x\right)$$

what is $$h'\left(x\right)$$ ?

• $$h'\left(x\right)=\frac{5}{3}x^{-2}-f'\left(x\right)$$

• $$h'\left(x\right)=-\frac{15}{x^4}-f'\left(x\right)$$

• $$h'\left(x\right)=-\frac{15}{x^2}-f'\left(x\right)$$

Math Response

$$h\left(x\right)=\frac{5}{x^3}-f\left(x\right)$$

Find $$h'\left(1\right)$$

Type answer here

Correct Answer: -14

Deg^°

Rad

Multiple Choice

If $$h\left(x\right)=f\left(x\right)\cdot g\left(x\right)$$

what is $$h'\left(x\right)$$ ?

• $$h'\left(x\right)=f'\left(x\right)\cdot g'\left(x\right)$$

• $$h'\left(x\right)=f'\left(x\right)\cdot g\left(x\right)+g'\left(x\right)\cdot f\left(x\right)$$

• $$h'\left(x\right)=\frac{\left(f'\left(x\right)\cdot g\left(x\right)-g'\left(x\right)\cdot f\left(x\right)\right)}{\left(g\left(x\right)\right)^2}$$

• That's impossible, we haven't learned that technique.

Math Response

Find the value of $$h'\left(-8\right)$$

Type answer here

Correct Answer: -8

Deg^°

Rad

• What is this the meaning of this limit?

• What is the original function, and the x-value

• What is the value of the limit?

Math Response

$$\lim_{x\rightarrow2}\left(\frac{\sqrt[]{3-x}-\left(1\right)}{x-2}\right)$$

What is the function being derived?

(type the function only don't write "f(x)="

Type answer here

Correct Answer: \sqrt{3-x}

Deg^°

Rad

Math Response

$$\lim_{x\rightarrow2}\left(\frac{\sqrt[]{3-x}-\left(1\right)}{x-2}\right)$$

What is the x-value of the derivative?

Type answer here

Correct Answer: 2

Deg^°

Rad

Math Response

$$f\left(x\right)=\sqrt[]{3-x}$$

What is the derivative of f(x)?

Type answer here

Correct Answer: -\frac{1}{2\sqrt{3-x}}

Deg^°

Rad

Math Response

$$f'\left(2\right)=\lim_{x\rightarrow2}\left(\frac{\sqrt[]{3-x}-\left(1\right)}{x-2}\right)$$

Evaluate the Limit

Type answer here

Correct Answer: -\frac{1}{2}

Deg^°

Rad

​Solution

Categorize

Options (4)

$$\lim_{x\rightarrow-1}g\left(x\right)=DNE$$  

g is not cont. at x=-1

$$\lim_{x\rightarrow-1}\left(\frac{g\left(x\right)-g\left(-1\right)}{x+1}\right)\ DNE$$  

g'(-1)=0

If a function, g, is NOT differentiable at x=-1, which of the following must be TRUE or False?

True

False

Math Response

$$\frac{d^2y}{dx^2}=\frac{1}{3}x^3+4x^2-5$$

What is the value of

$$\frac{d^4y}{dx^4}$$

Type answer here

Correct Answer: 2x+8

Deg^°

Rad

Math Response

$$\frac{d^2y}{dx^2}=\frac{1}{3}x^3+4x^2-5$$

What is the value of

$$\frac{d^4y}{dx^4}$$ at x=-2

Type answer here

Correct Answer: 4

Deg^°

Rad

Multiple Choice

$$f\left(x\right)=3x^4-5$$ and g is differentiable.

Find the derivative of $$f\left(g\left(x\right)\right)$$

• $$3\left(g'\left(x\right)\right)^3$$

• $$12x^3\left(g'\left(x\right)\right)$$

• $$12\left(g\left(x\right)\right)^3\left(g'\left(x\right)\right)$$

• $$12\left(g'\left(x\right)\right)^3-5$$