Question

(a) Suppose f and g are di?erentiable on an interval I and that f(x) − ((g(x))n = c for all

xεI (where nεNand cεR are constants). If g(x) ̸= 0 on I, then g′(x) = −f(x)((g(x))1−n.n

(b) If f is not di?erentiable at x0,then f is not continuous at x0.

(c) Suppose f and g are di?erentiable on an interval I and suppose that f′(x) = g′(x)on I.

Then f(x) = g(x) on I.

(d) The equation of the line tangent to y = f(x) at the point (a, f(a)) is given by y − f(a) =(f′(x))(x − a).

(e) Iff(0)=0,thenf′(0)=0.

Find the indicated derative

d^99/dx^99.e^2x

d^n/dx^n[1/1-x]

Answer #1

Please write question number 1 properly again.

13. Consider
f(x)=sqrt x-2
a) Using any of the three limit formulas to find f ′ ( a
), what is the slope of the tangent line to f ( x )at x = 18? (6
points execution, 2 points notation)
b) Find the equation of the tangent line at x =
18
14. State the derivative.
a) d/ d x [ x ^n ]
b) d /d x [ cos x ]
c) d /d x [ csc...

Suppose that f(x) = (2x)/((4-2x)^3)
Find an equation for the tangent line to the graph of f at
x=1.
Tangent line: y =

Need work shown, please
a) Suppose a function f is continuous on the interval [a,b]. If
f(a) is negative and f(b) is positive, explain why there must be a
number c between a and b such that f(c) = 0.
(Context: related to “Intermediate Value Theorem”.)
b) Use the idea from part (a) to show that the equation x^5 = 2
− 2x has a solution in the interval [0, 1].
(Hint: A solution for an equation f(x) = g(x)...

Let f(x)= a -bx^c + dx^e where a, b,c,d,e >0 and c<e.
Suppose that f(x0)= 0 and f '(x0)=0 for some x0>0. Prove that
f(x) greater than or equal to 0 for x greater than or equal to
0

Define g(x) = f(x) + tan-1 (2x) on [−1, √3/2 ].
Suppose that both f'' and g'' are continuous for all x-values on
[−1, √3/2 ]. Suppose that the only local extrema that f has on the
interval [−1, √3/2 ] is a local minimum at x = 1/2 .
(a) Determine the open intervals of increasing and decreasing
for g on the interval [1/2 , √3/2] .
(b) Suppose f(1/2) = 0 and f(√3/2) = 2. Find the absolute...

Suppose the derivative of f exists, and assume that f(1) = 4,
and f'(1) = 5. Let g(x) = x^2f(x), and h(x) = f(x)/x-2
a) g' (1) = ??
find the equation of the tangent line to g(x) at x = 1
y = ??
b) h'(1) = ??
Find the equation of the tangent line to h(x) at x = 1
y = ??

a) Find the equation of the tangent line to f(x) = e –x at the
point (1, e –1).
b) Find the equation of the tangent line to f(x) = 3x2 – 2x + 5 at
x = 2.

Suppose f(x) = (2/x) + 5 .
a. *Graph this function.
b. *Find the equation of the secant line to f(x) on the interval
[1, 3]. Call this line g(x). Add g(x) to your graph
c. Find the equation of the tangent line to f(x) at the point
(2,6). Call this line h(x). Add h(x) to your graph.
Please neatly show your work.

Suppose that f is a bijection and f ∘ g is defined. Prove:
(i). g is an injection iff f ∘ g is;
(ii). g is a surjection iff f ∘ g is.

The table below shows values for four differentiable functions.
Suppose we know the following things:
h'(x) = g(x)
g'(x) = f(x)
f'(x) = b(x)
b'(x) = h(x)
0
1
2
3
4
b(x)
2
4
3
1
0
f(x)
1
4
2
3
0
h(x)
2
0
3
1
4
g(x)
2
0
4
3
1
a) What is intergal from a = 2 and b = 4 f(x) dx?
b) What is intergal from a = 0 and b =...

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