Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown.

Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown.
(a) Find P ' (2)
(b) Find Q ' (7)

Solutions

Expert Solution

(a) P'(2) will be the derivative of P(x) evaluated at x=2. So first, we take a derivative.

In this section, you should have learned the product rule, so P'(x) will look like this:

P'(x)=F(x)G'(x)+F'(x)G(x)

Now we let x=2.

P'(2)=F(2)G'(2)+F'(2)G(2)

Read these values of the graph.
We see that F(2)=3. (When its x-value is 2, the curve F has a y-value of 3)
We see that G(2)=2.
F'(2) will be the slope of the curve F at 2. A tangent line at F(2) would be horizontal, so F'(2)=0.
G(2) has a slope of 1/2, so G'(2)=1/2

Now we substitute these values into P'(2) to solve:

P'(2)=(3)(1/2)+(0)(2)=(3/2)+(0)=3/2

(b) We're playing by the same rules as above--differentiate the equation, read the values of the graph, substitute the values into the function, and solve.

Differentiate (Quotient Rule):

Q'(x)=[G(x)F'(x)-F(x)G'(x)]/[G(x)^2]

Read the values of the graph:

G(7)=1

G'(7)=-2/3

F(7)=5

F'(7)=1/4

Substitute:

Q'(7)=[1*(1/4)-5*(-2/3)]/[1^2]

Q'(7)=[(1/4)+(10/3)]

Q'(7)=(43/12)=3.5833

Dr. OWL answered 3 years ago

Next > < Previous

Related Solutions

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x).

if f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x) (a) Find u'(1) (b) Find v'(5).

Let f and g be two functions whose ﬁrst and second order derivative functions are continuous,...

Let f and g be two functions whose ﬁrst and second order derivative functions are continuous, all deﬁned on R. What assumptions on f and g guarantee that the composite function f ◦g is concave?

Let p, q, g : R → R be continuous functions. Let L[y] := y'' +...

Let p, q, g : R → R be continuous functions. Let L[y] := y'' + py' + qy. (i) Explain what it means for a pair of functions y1 and y2 to be a fundamental solution set for the equation L[y] = 0. (ii) State a theorem detailing the general solution of the differential equation L[y] = g(t) in terms of solutions to this, and a related, equation.

Let p and q be propositions. (i) Show (p →q) ≡ (p ∧ ¬q) →F (ii.)...

Let p and q be propositions. (i) Show (p →q) ≡ (p ∧ ¬q) →F (ii.) Why does this equivalency allow us to use the proof by contradiction technique?

3. Let F : X → Y and G: Y → Z be functions. i. If...

3. Let F : X → Y and G: Y → Z be functions. i. If G ◦ F is injective, then F is injective. ii. If G ◦ F is surjective, then G is surjective. iii. If G ◦ F is constant, then F is constant or G is constant. iv. If F is constant or G is constant, then G ◦ F is constant.

Let f : X → Y and g : Y → Z be functions. We can...

Let f : X → Y and g : Y → Z be functions. We can deﬁne the composition of f and g to be the function g◦ f : X → Z for which the image of each x ∈ X is g(f(x)). That is, plug x into f, then plug theresultinto g (justlikecompositioninalgebraandcalculus). (a) If f and g arebothinjective,must g◦ f beinjective? Explain. (b) If f and g arebothsurjective,must g◦ f besurjective? Explain. (c) Suppose g◦ f isinjective....

Let f(x) and g(x) be two generic functions. Assume limx→0(f(x)+2g(x))=2 & limx→0(f(x)−g(x))=8. Compute limx→1(f(lnx)/g(x2−x)). A. It...

Let f(x) and g(x) be two generic functions. Assume limx→0(f(x)+2g(x))=2 & limx→0(f(x)−g(x))=8. Compute limx→1(f(lnx)/g(x2−x)). A. It cannot be computed B. 2 C. 4 D. -3 E. -4

Let f(x) = 5x+3 and g(x) =2x-5. Find (f+g)(x),(f-g)(x),(fg)(x), and (f/g) (x). Give the domain of...

Let f(x) = 5x+3 and g(x) =2x-5. Find (f+g)(x),(f-g)(x),(fg)(x), and (f/g) (x). Give the domain of each. (f+g) (x) = (f-g)(x) = (fg)(x) = (f/g)(x) = The domain of f+g is_ The domain of f-g is_ The domain of fg is _ The domain of f/g is _ Please at the end provide showed work.

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore,...

Let⇀F and⇀G be vector fields defined on R3 whose component functions have continuous partial derivatives. Furthermore, assume that ⇀∇×⇀F=⇀∇×⇀G.Show that there is a scalar function f such that ⇀G=⇀F+⇀∇f.

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P Q=S(P,t) ∂S>0, ∂S<0 ∂P ∂t where t is the tax rate on the commodity. (a) What are the endogenous and exogenous variables? (b) Derive the total differential of each equation. (c) Use Cramer’s rule to compute dQ/dt and dP/dt. (d) Determine the sign of dQ/dt and dP/dt. (e) Use the Q − P diagram to explain your results. Find the Taylor series with n...

Subjects