Victoria, Pavi, Susan
1)
If f(2) = 3 and f ’(2) = 5, find an equation of (a) the tangent line, and (b) the normal
line to the graph of y = f(x).
2)
On the same coordinate plane, graph the derivative of the function y = f(x) shown.
At what values of x within the domain is the function not differentiable?
3) List four situations in which a graph would NOT be differentiable.
4)
Find derivative of y = x4 - 4x3+1.
5) Find instantaneous rate of change of the area A with respect to the radius r of the circle.
6)
𝑓(𝑥) = sec 𝑥
a. Sketch the graph (hint: graph cos x first)
𝜋
b. Find sec( ).
4
𝜋
c. at ( , √2), find the tangent line.
4
d. Find the normal line.
7)
Using the Chain Rule, find dy/dx of y = sin-5x – cos3x.
8)
Given a parametrically defined function x = t, y = t, and t = 14, find the equation
of the line tangent to the curve.
9)
𝑥 2 − 𝑥𝑦 + 𝑦 2 = 7
Find 𝑓′(𝑥) using Implicit Differentiation.
10)
Find the derivative of y=s√1 − 𝑠 2 + cos-1s.
11)
Which of the following gives dy/dx if y = log (2x – 3)
10
A.
D.
12)
2
(2𝑥−3) ln 10
1
2𝑥−3
2
B.
E.
2𝑥−3
1
2𝑥
Find the derivative of y = 3cotx.
ANSWER KEY
C.
1
(2𝑥−3) ln 10
1) If f(2) = 3 and f ’(2) = 5, find an equation of (a) the tangent line, and (b) the normal line
to the graph of y = f(x).
Given that f(2) = 3, a line passes through the point (2,3).
Given that f ’(2) = 5, the derivative of the graph is 5 anywhere on that line.
Substitute into point slope form:
(a) tangent line: y – 3 = 5(x – 2)
(b) normal line : y – 3 = - 15(x – 2)
2)
Graph the derivative of the function y = f(x) shown.
At what values of x within the domain is the function not differentiable?
x = 0, x = 1, and x = 4
3) List four situations in which a graph would NOT be differentiable.
A graph is not differentiable where there is cusp, a corner, a vertical tangent line, and a
discontinuity.
4) Find derivative of y = x4 - 4x3+1.
y’ = 4x3 – 12x2
Use the power rule.
5) Find instantaneous rate of change of the area A with respect to the radius r of the circle.
A’ = 4r
The instantaneous rate of change refers to the derivative; therefore, find the
derivative of A.
6) 𝑓 (𝑥) = sec 𝑥
a. Sketch the graph (hint: graph cos x first)
𝜋
1
4
cos⁡( )
b. sec( ) =
𝜋
4
=
1
√2
2
=
2
√2
= √2
𝜋
𝜋
c. at ( , √2), find tangent line: 𝑦 − √2 = ⁡ √2(𝑥 − )
4
4
d. normal line: 𝑦 − √2 = ⁡
−1
√2
𝜋
(𝑥 − )
4
7) Using the Chain Rule, find the derivative of y = sin x – cos x.
y = sin x – cos x = (sinx) – (cosx)
y’= -5(sinx) (cosx) – 3(cosx) (-sinx)
y’ = -5sin xcosx + 3cos xsinx
-5
-5
3
-5
-6
-6
2
2
3
3
8) Given a parametrically defined function x = t, y = t, and t = 14, find the equation of
the line tangent to the curve.
Find derivative of each part of the defined function: x' = 1
1
1
y’ = t1/2 = t -1/2 =
2
Derivative of parametric curve:
𝑦′
𝑥′
Substitute value of t into derivative:
=
1
2√𝑡
1
1
1
2√4
=
1
=
2√ 𝑡
1
1
2(2)
Find value of x and y by substituting t into each:
Equation of tangent line:
9)
y-
1
2
1
= (x - )
4
𝑥 2 − 𝑥𝑦 + 𝑦 2 = 7
Find 𝑓′(𝑥) using Implicit Differentiation.
2x – [(xy’+ y)] + 2yy’ = 0
2x – xy’ – y + 2yy’ = 0
-xy’ + 2yy’ = -2x + y
(y’)(-x + 2y) = -2x + y
y' =
10)
2√𝑡
−2𝑥+𝑦
2𝑦−𝑥
Find the derivative of y=s√1 − 𝑠 2 + cos-1s.
=1
x=
1
4
1
1
4
2
y=√ =
11)
Which of the following gives dy/dx if y = log10(2x – 3)?
2
Answer: A. (2𝑥−3)
ln 10
Using definition for derivative of a log function,
𝑑
𝑑𝑥
1
(log10 (2𝑥 − 3)) = (2𝑥−3)ln10 ∙ (2x – 3)’
1
= (2𝑥−3)ln10 ∙ 2
=
12)
Find the derivative of y = 3cotx.
Using definition for derivative of a , y’ = ln3 ∙ 3cotx ∙ (cotx)’
= ln3 ∙ 3cotx ∙ (-cscx)
= -3cotxln3(cscx)
x
2
(2𝑥−3)ln10