AP Calculus AB - Important Formulas/Theorems

Which of these is the definition of a derivative?

• $\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)-f\left(x\right)}{h}$

• $\lim_{h\rightarrow0}\ \frac{f\left(h\right)-f\left(x\right)}{h}$

• $\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)+f\left(x\right)}{h}$

• $\lim_{h\rightarrow0}\ \frac{f\left(h\right)+f\left(x\right)}{h}$

The intermediate value theorem (IVT) is primarily concerned with which of the following?

• y-values

• first derivative values

• second derivative values

• x-values

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

Which of these sums up the Mean Value Theorem (MVT)?

• $f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

• $f\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

• $f\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

• $f'\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

Tags

CCSS.HSF.IF.B.6

Which of these is NOT a hypothesis of the Mean Value Theorem (MVT)?

• A closed interval

• A differentiable function

• A continuous function

• A twice-differentiable function

Tangent line formula.

• $y-f\left(x_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

• $y=f'\left(x_1\right)\left(x-x_1\right)$

• $y-f\left(y_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

• $y=f\left(x_1\right)\left(x-x_1\right)$

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

The fundamental theorem of calculus.

• $\int_a^bf\left(x\right)=F\left(b\right)-F\left(a\right)\$ where F is the antiderivative

• $\int_a^bf\left(x\right)=f'\left(b\right)-f'\left(a\right)\$

• $\int_a^bF\left(x\right)=f\left(b\right)-f\left(a\right)\$ where F is the antiderivative

• $\int_a^bF\left(x\right)=f'\left(b\right)-f'\left(a\right)\$ where F is the antiderivative

Tags

CCSS.HSF.IF.A.2

The fundamental theorem of calculus.

• $\frac{d}{dx}\int_0^xf\left(t\right)dt=f\left(x\right)$

• $\frac{d}{dx}\int_0^xf\left(t\right)dt=F\left(x\right)$ where F is the antiderivative

• $\frac{d}{dx}\int_0^xF\left(t\right)dt=f\left(x\right)$ where F is the antiderivative

• $\frac{d}{dx}\int_0^xf\left(t\right)dt=f'\left(x\right)$

Tags

CCSS.HSF.IF.C.7

Volume using discs revolving around horizontal line. 

•  $\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$  

•  $\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$  

•  $\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$  

•  $\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$  

Tags

CCSS.HSG.GMD.A.3

Volume using discs revolving around vertical line. 

•  $\pi\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$  

•  $\pi\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$  

•  $\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$  

•  $\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$  

Tags

CCSS.HSG.GMD.A.1

CCSS.HSG.GMD.A.2

CCSS.HSG.GMD.A.3

Volume using washers revolving around horizontal line. 

•  $\pi\int_{x=a}^{x=b}R^2-r^2\ dx\ \$  

•  $\pi\int_{x=a}^{x=b}\left(R-r\right)^{2\ }dx\ \$  

•  $\int_{x=a}^{x=b}\left(R-r\right)^2dx\ \$  

•  $\int_{x=a}^{x=b}R^{2\ }-r^{2\ }dx\ \$  

Tags

CCSS.HSG.GMD.A.2

CCSS.HSG.GMD.A.3