Transcript Semester Exam Review - Campbell County High School

Semester Exam
Review
{
AP Calculus
Exam Topics
Limits (algebraically & by graph)
 Find derivatives using limit definition
 Given a graph, sketch derivative graph
 Derivatives

Power Rule
 Chain Rule
 Product/Quotient Rules
𝑥
𝑥
 𝑒 , 2 , ln x
 Trig derivatives
 Inverse trig derivatives

Indefinite Integrals (+ c !!!)
 Definite Integrals (Fundamental Theorem)

𝑠𝑖𝑛′ 𝑥 = cos 𝑥
𝑠𝑒𝑐 ′ 𝑥 = sec 𝑥 tan 𝑥
tan′ 𝑥 = 𝑠𝑒𝑐 2 𝑥
𝑐𝑜𝑠 ′ 𝑥 = −𝑠𝑖𝑛𝑥
𝑐𝑠𝑐 ′ 𝑥 = − csc 𝑥𝑐𝑜𝑡 𝑥
𝑐𝑜𝑡 ′ 𝑥 = −𝑐𝑠𝑐 2 𝑥
Trig function derivatives
Differentiate:
𝑑
sin−1 𝑥 =
𝑑𝑥
𝑑
sec −1 𝑥
𝑑𝑥
=
1
1 − 𝑥2
1
|𝑥| 𝑥 2 −1
𝑑
1
−1
tan 𝑥 = 2
𝑑𝑥
𝑥 +1
Co-Functions: Negative!!
𝑑
1
−1
csc 𝑥 = −
𝑑𝑥
|𝑥| 𝑥 2 − 1
Find f’(x) using the limit
definition of derivatives:
𝑓 𝑥 = 3𝑥 2 − 4𝑥 + 7
𝑓 𝑥+ℎ −𝑓(𝑥)
lim
ℎ
ℎ→0
“forward difference quotient”
Derivatives by limits:
𝑓 𝑥 + ℎ − 𝑓(𝑥)
lim
ℎ→0
ℎ
3(𝑥 + ℎ)2 −4 𝑥 + ℎ + 7 − (3𝑥 2 − 4𝑥 + 7)
= lim
ℎ→0
ℎ
=
3𝑥 2 +6𝑥ℎ+3ℎ2 −4𝑥−4ℎ+7−3𝑥 2 +4𝑥−7
lim
ℎ
ℎ→0
= lim 6𝑥 + 3ℎ − 4 = 6𝑥 − 4
ℎ→0
Rate of Change:
Average vs. Instantaneous
Average Rate of Change – NOT AN AVERAGE!!!
(slope of secant!):
∆𝒀
∆𝑿
=
𝒇 𝒃 −𝒇(𝒂)
𝒃−𝒂
Units: mi/hr, ft/s, etc.
𝑡𝑜𝑡𝑎𝑙 𝑚𝑖𝑙𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 ℎ𝑜𝑢𝑟𝑠
Instantaneous Rate of Change =
DERIVATIVE!!!!
(Slope of tangent line)
Writing Equation of Tangent
Line
Write the equation of the tangent line to
f(x) at x = 2 if 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝟕
-- 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)
Displacement function:
𝑑 𝑡 = 5𝑡 3 − 7𝑡 + 1
Derivative of displacement is velocity:
𝑣 𝑡 = 15𝑡 2 − 7
Derivative of velocity is acceleration:
a(t) = 30t
Displacement/Velocity/Acceleration
Implicit Differentiation:
2
4𝑥 + tan 𝑥𝑦 = 𝑦
3
Differentiate implicitly:
2
4𝑥 + tan 𝑥𝑦 = 𝑦
3
Derivative:
𝑑𝑦
8𝑥 + 𝑦𝑠𝑒𝑐 2 𝑥𝑦
=
𝑑𝑥
3𝑦 2 − 𝑥𝑠𝑒𝑐 2 𝑥𝑦
To the nearest thousandth, calculate
the slope of the tangent where x = 4:
𝑥 2 − 4𝑦 2 = 4
To the nearest thousandth, calculate
the slope of the tangent where x = 4:
𝑥 2 − 4𝑦 2 = 4
Differentiate implicitly:
Find coordinates of y when x = 4
and substitute into dy/dx equation:
𝑑𝑥
𝑑𝑦
2𝑥 ∙
− 8y ∙
=0
𝑑𝑥
𝑑𝑥
𝑑𝑦
= −8y ∙
= −2𝑥
𝑑𝑥
𝑑𝑦 −2𝑥 𝑑𝑦
𝑥
= −8y ∙
=
→
=
𝑑𝑥 −8𝑦 𝑑𝑥
4𝑦
𝑊ℎ𝑒𝑛 𝑥 = 4, 𝑦 = ± 3
𝑑𝑦
4
=
≈ .577
𝑑𝑥 4( 3)
𝑑𝑦
4
=
≈ −.577
𝑑𝑥 4(− 3)
Useful Related Rates Formulas
𝐶𝑖𝑟𝑐𝑙𝑒 𝐴𝑟𝑒𝑎: 𝜋𝑟 2
1 2
𝐶𝑜𝑛𝑒 𝑉𝑜𝑙𝑢𝑚𝑒: 𝑉 = 𝜋𝑟 ℎ
3
Cylinder Volume: V = 𝜋𝑟 2 ℎ
𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑇ℎ𝑒𝑜𝑟𝑒𝑚: 𝑎2 + 𝑏 2 = 𝑐 2
𝑆𝑖𝑚𝑖𝑙𝑎𝑟 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 (𝑆𝑒𝑡 𝑢𝑝 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛)
Differentiability Implies
Continuity
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!
To Prove Continuity:
Function f is continuous at x = c if and only if:
1.) f(c) exists
2.) lim 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠
𝑥→𝑐
3.) lim 𝑓 𝑥 = 𝑓(𝑐)
𝑥→𝑐
Trapezoidal
Rule/Riemann Sums
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:
1
ℎ(𝑏1
2
+ 𝑏2 )
Exam Topics Checklist






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!)