Semester Exam Review

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Semester Exam Review

AP Calculus

Limits (algebraically & by graph) Find derivatives using limit definition Given a graph, sketch derivative graph Derivatives

Power Rule Chain Rule Product/Quotient Rules , , ln x Trig derivatives Inverse trig derivatives

Indefinite Integrals (+ c !!!) Definite Integrals (Fundamental Theorem)

Exam Topics

Trig function derivatives

Differentiate:𝑑𝑑𝑥 sin

−1𝑥=1

√1−𝑥2

𝑑𝑑𝑥 csc

−1𝑥=− 1¿𝑥∨√𝑥2−1

𝑑𝑑𝑥 tan

− 1𝑥=1

𝑥2+1 Co-Functions: Negative!!

“forward difference quotient”

Find f’(x) using the limit definition of derivatives:

Derivatives by limits:

=

=

Average Rate of Change – NOT AN AVERAGE!!!(slope of secant!):

Rate of Change:Average vs. Instantaneous

Units: mi/hr, ft/s, etc.

DERIVATIVE!!!!(Slope of tangent line)

Instantaneous Rate of Change =

Write the equation of the tangent line to f(x) at x = 2 if

-- Use slope-intercept form: y – y1 = m(x – x1)f(2) = 19, so (2, 19) is a point on graph

-- Use derivative to find slope of tan. at x = 2.

f’(x) = 6x 6(2) = 12

y – 19 = 12(x – 2) (can write in slope-int form as well)

Writing Equation of Tangent Line

Displacement function:Derivative of displacement is velocity:Derivative of velocity is acceleration:

a(t) = 30t

Displacement/Velocity/Acceleration

Implicit Differentiation:

4 𝑥2+ tan 𝑥𝑦=𝑦3

Differentiate implicitly:

Derivative:

To the nearest thousandth, calculate the slope of the tangent where x = 4:

𝑥2−4 𝑦2=4

To the nearest thousandth, calculate the slope of the tangent where x = 4:

𝑥2−4 𝑦2=4Differentiate implicitly:

2 𝑥 ∙ 𝑑𝑥𝑑𝑥 −8 y ∙𝑑𝑦𝑑𝑥=0

¿−8 y ∙𝑑𝑦𝑑𝑥=−2 𝑥

¿−8 y ∙𝑑𝑦𝑑𝑥=−2𝑥−8 𝑦→

𝑑𝑦𝑑𝑥=

𝑥4 𝑦

Find coordinates of y when x = 4and substitute into dy/dx equation:

h𝑊 𝑒𝑛𝑥=4 , 𝑦=±√3𝑑𝑦𝑑𝑥 =

44¿¿

𝑑𝑦𝑑𝑥 =

44¿¿

Useful Related Rates Formulas

𝐶𝑜𝑛𝑒𝑉𝑜𝑙𝑢𝑚𝑒 :𝑉=13 𝜋 𝑟

2h

𝐶𝑖𝑟𝑐𝑙𝑒 𝐴𝑟𝑒𝑎 :𝜋𝑟2

Cylinder Volume: V =

h𝑃𝑦𝑡 𝑎𝑔𝑜𝑟𝑒𝑎𝑛 h𝑇 𝑒𝑜𝑟𝑒𝑚 :𝑎2+𝑏2=𝑐2

𝑆𝑖𝑚𝑖𝑙𝑎𝑟 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 (𝑆𝑒𝑡 𝑢𝑝𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛)

Be able to find values that make a piecewise function differentiable at a given point (must be continuous AND differentiable)

Remember to use LIMITS to show differentiability and continuity!

Differentiability Implies Continuity

Function f is continuous at x = c if and only if:1.) f(c) exists2.) 3.)

To Prove Continuity:

Make sure to draw graph! Check on calculator when possible, but show all setup

Remember that the AP exam tends to use uneven intervals so you have to do it by hand

Trapezoid area:

Trapezoidal Rule/Riemann Sums

Implicit Differentiation Differential dy Average Rate of Change Instantaneous Rate of Change Estimate definite integrals using trap rule, Riemann sums, graph Applications

Find velocity given displacement equation Find displacement given velocity equation Write equation of tangent line Find c value guaranteed by Mean Value Theorem Related Rates

Calculator Find numerical derivatives Table of Values/Graph Riemann Sums/Trapezoidal Rule Program (especially for large values of

n!)

Exam Topics Checklist