A research probe is submerged in a fluid where the vertical force, F F\,F kilonewtons, exerted on its hull is modeled by the function
F(d)=54d2+4d−26,d>0 F(d) = \frac{54}{d^2} + 4d - 26, \quad d > 0 F(d)=d254+4d−26,d>0where d d\,d is the depth in metres below the surface.
Using calculus,
determine the range of depths for which the vertical force F(d)F(d)F(d) is increasing.
show that ∫39(54d2+4d−26)dd=0\displaystyle \int_{3}^{9} \left( \frac{54}{d^2} + 4d - 26 \right) dd = 0∫39(d254+4d−26)dd=0.
The points A(3,−8)A(3, -8)A(3,−8) and B(6,−0.5)B(6, -0.5)B(6,−0.5) lie on the curve F(d)F(d)F(d).
Given that ∫36(54d2+4d−26)dd=−15\displaystyle \int_{3}^{6} \left( \frac{54}{d^2} + 4d - 26 \right) dd = -15∫36(d254+4d−26)dd=−15.
(i) state the value of ∫69(54d2+4d−26)dd\displaystyle \int_{6}^{9} \left( \frac{54}{d^2} + 4d - 26 \right) dd∫69(d254+4d−26)dd.
(ii) find the value of the constant k k\,k such that ∫36(54d2+4d+k)dd=0\displaystyle \int_{3}^{6} \left( \frac{54}{d^2} + 4d + k \right) dd = 0∫36(d254+4d+k)dd=0.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.