Chris Seo, Nathaniel Wichman, Sean Moore,Tyler Pritchard
IB Calculus
Mr. Wahbeh
Chapter 3 Practice Quiz
3.1 - Derivative of a Function
Find the derivative of the given function at the indicated point for the following
questions:
1. f(x)=2x, a=2
2. f(x)=x2+4, a=1
3. f(x)=3-x2, a=-1
4. f(x)=x2+x, a=0
Find...
1. ddx(x3)
2. ddx(x3+3x2+4)
3. ddxf(x) if f(x)=4x4
4. ddx(12x5+45x3+❤)
3.2 - Differentiability
True or False
1. If f has a derivative at x=a, then f is continuous at x=a.
2. If f is continuous at x=a, then f has a derivative at x=a.
The function fails to be differentiable at x=0. Tell whether it is a corner, a usp, a
vertical tangent, or a discontinuity
1. y=x4/5
2. y=3-3x
3.3 - Rules of Differentiation
Find the dy/dx
1. y=-x3+4
2. y=x2+x+1
3. y=x33-x
Find the horizontal tangents of the curve
1. y=x3+x2-x
2. x3-2x2+x+1
Find the equation for the line tangent to the curve at the given point
1. y=x3+12x, x=1
3.4 - Velocity and Other Rates of Change
1. the figure below depicts the velocity v = f(t) of a dangerous, out of control
freight train
a. when is the train moving backwards?
b. when is the train speeding up?
c. when does the train have a negative acceleration?
d. when does the train change directions?
2.
find the instantaneous velocity at t=4 of a dangerous, out of control train
whose position is given by the function s(t) = t2 – 3t + 2, where s is measured in
kilometers and t is measured in nanoseconds
3.5 - Derivatives of Trigonometric Functions
1. find the y’, y’’, y’’’ of y=sin(x)
2. find the derivative of...
a.
tan(5x)
b.
sec(3x)
c.
cot(7x)
d.
csc(9x)
3.6 - Chain Rule
Use the chain rule to find the derivatives of the following equations.
1. y = sin (3x +1)
2. y = cos (x3)
3. y = sin3x tan 4x
4. y = 32x + 1
Find the tangent line of (f o g) at the given value of x.
1. f(x)= x5+ 1, g(x)=x, x=1
2. f(x)=cotx10, g(x)=5x, x=1
Multiple Choice
1. Which of the following is dy/dx if y=tan(4x)?
a.
4 sec (4x) tan (4x)
b.
4 sec2(4x)
c.
4 cot (4x)
2.
Which of the following is y’ if y=cos2(x3+x2)
a.
-2(3x2+2x) cos (x3+x2) sin (x3+x2)
b.
2(3x2+2x) cos (x3+x2) sin (x3+x2)
c.
-2(3x2+2x)
3.7 - Implicit Differentiation
Find the derivatives
1. x2y – xy2 = 6
2. x + tan (xy) = 0
Find the equation of lines that are (a) tangent and (b) normal to the curve at the
given point
3.
x2y2 = 9, (-1,3)
3.8 - Derivatives of Inverse Trigonometric Function
Use the derivatives of inverse trig. functions to find the value of y’.
1. y=cos-1(x2)
2. y=csc-1(x2+1)
3. y=sin-1(3x2)
4. y=cot-1x-1
Find the tangent line for the graph of y at the following indicated point.
1. y=sin-1(x4), x=2
2. y=tan-1(x2), x=1
Multiple Choice
1. Which of the following is the derivative of y=tan-1(3x)
a.
y'=-31+9x2
b.
y'=31-9x2
c.
y'=31+9x2
2.
Which of the following is the derivative of y=sec-1(x2)
a.
y'=2x1-x4
b.
y'=2x2x4-1
c.
y'=2x2x2-1
3.9 - Derivatives of Exponential and Logarithmic Function
1. What is the derivative e❤?
a.
e❤❤
b.
e❤❤’
c.
1❤❤’
2. What is the derivative of ❤❤?
.
❤❤ln(❤+1)
a.
❤❤log(❤+1)
b.
❤❤log(❤)
3. Find the derivatives of the following?
.
y= ln(6x+9)
a.
y= log6(3x2)
Answer Key
3.1
Find the derivative of the given function
1. 1/2
2. 2
3. 2
4. 1
Find...
1. 3x2
2. 3x2+6x
3. 16x3
4. 60x4+135x2
3.2
True or False
1. True
2. False
Tell whether the problem is a corner, a cusp, a vertical tangent, or a discontinuity
1. Cusp
2. Vertical Tangent
3.3
Find dy/dx
1. -3x
2. 2x+1
3. x2-1
Find the horizontal tangents of the curve
1. x=13, x=-1
2. x=1, x=13
Find the equation for the line tangent to the curve at the given point
1. y-1=12(x-1)
3.4
1. velocity of Thomas the Tank Engine
a.
moving backwards at 1<t<5
b.
speeds up at 1<t<2 and 5<t<6
c.
has a negative acceleration at 0≤t<2 and 6<t<7
d.
3.5
changes direction at t=1 and t=5
2. 5 km/ns
1. y’=cos(x)
y’’=-sin(x)
y’’’=-cos(x)
2.
a. 5sec2(5x)
b. 3sec(3x)tan(3x)
c.-7csc2(7x)
d.-9csc(9x)cot(9x)
3.6
Finding derivatives
1. y'=3 cos (3x+1)
2. y'=-3 sin (x3)
3. y'=4 sin3x sec24x+3 sin2x cos x tan 4x
4. y'=-3(2x+1)-32
Finding the tangent line of (f o g)
1. y-2=52(x-1)
2. y-2=-4(x-1)
Multiple Choice
1. B
2. A
3.7
Finding derivatives
1. y’ =-2xy + y22xy + x2
2. y’ = -1xcos2(xy)-yx
3.
a.
y = 3x + 6
b.
y = -13x+83
3.8
Finding Derivatives
1. y'=-2x1-x4
2. y'=-2(x2+1)x2+2
3. y'=-6xx4-9
4. y'=-12xx-1
Finding the tangent of the line
1. y=0.378x+0.286
2. y=0.118x+0.668
Multiple Choice
1. C
2. B
3.9
1. B
2. A
3. a. y’=63x+9
b. y’=6xln(6)3x